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THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
CONDUCTED BY
LORD KELVIN, G.C.V.O. D.C.L. LL.D. F.R.S. &c.
GEORGE FRANCIS FITZGERALD, M.A. Sc.D. F.R.S.
AND
WILLIAM FRANCIS, Ph.D. F.L.S. F.R.A.S. F.C.S.
11 Nee aranearum sane textus ideo melior quia ex se fila gignunt, nee noster
vilior quia ex alienis libaraus ut apes." Just. Lips. Polit. lib. i. cap. 1. Not.
VOL. XLVIL— FIFTH SERIES.
JANUARY— JUNE 1899.
LONDON:
TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET.
SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD. ; tVHITTAKEB AND CO.
AND BY ADAM AND CHARLES BLACK; T. AND T.CLARK, EDINBURGH;
SMITH AND SON, GLASGOW; — HODGES, FIGGIS, AND CO , DUBLIN J
PUTNAM, NEW YORE, — VEUVE J. BOYVEAU; PARIS \—
AND ASHER AND CO,, BERLIN.
f*1 O /"* O i*"
 . » ;. >
•/
" Meditation is est perscrutari occulta; contemplationis est admirari
perspicua .... Adniiratio generat quaestionem, quaestio iuvestigationem,
investigatio inventionem." — Hugo de 8. Victore,
" Cur spirent venti, cur terra dehiscat,
Cur mare turgescat, pelago cur tantus amaror,
Cur caput obscura Phoebus ferrugine condat,
Quid toties diros cogat flagrare conietas,
Quid pariat nubes, yeniant cur fuluiina coelo,
Quo micet igne Iris, superos quis conciat orbes
Tam vario motu."
J. B. Tinelli ad Mazonium.
FLAJIMAM.
CONTENTS OF VOL. XLVII
(FIFTH SEEIES).
NUMBEE CCLXXXIV.— JANUAEY 1899.
 Page
Mr. Albert Campbell on the Magnetic Fluxes in Meters and
other Electrical Instruments 1
Messrs. Edward B. Eosa and Arthur W. Smith on a Eeson
ance Method of Measuring Energy dissipated in Con
densers 19
Dr. E. H. Cook on Experiments with the Brush Discharge.
(Plate I.) 40
Dr. van Eijckevorsel on the Analogy of some Irregularities
in the Yearly Range of Meteorological and Magnetic
Phenomena. (Plate II.) 57
Lord Kelvin on the Age of the Earth as an Abode fitted for
Life 66
Mr. D. L. Chapman on the Eate of Explosion in Gases .... 90
Prof. Carl Bar us on the Aqueous Fusion of Glass, its Eelation
to Pressure and Temperature 104
Prof. E. Eutherford on Uranium Eadiation and the Electrical
Conduction produced by it 109
Notices respecting New Books : —
Dr. E. J. Eouth's Treatise on Dynamics of a Particle . . 163
Susceptibility of Diamagnetic and Weakly Magnetic Sub
stances, by A. P. Wills 164
NUMBER CCLXXXV.— FEBRUARY.
Dr. Thomas Preston on Eadiation Phenomena in the Mag
netic Field. — Magnetic Perturbations of the Spectral
Lines 165
IV CONTENTS OF VOL. XLVII. FIFTH SERIES.
Page
Lord Kelvin on the .Reflexion and Refraction of Solitary Plane
Waves at a Plane Interface between two Isotropic Elastic
Medinms— Fluid, Solid, or Ether 179
Prof. H. L. CalleDdar : Notes on Platinum Thermometry . . 191
Messrs. Edward B. Rosa and Arthur W. Smith on a Calori
metric Determination of Energy dissipated in Condensers . 222
Prof. Karl Pearson on certain Properties of the Hyper
geometrical Series, and on the fitting of such Series to
Observation Polygons in the Theory ot Chance 236
Lord Rayleigh on James Bernoulli's Theorem in Probabilities . 246
Notices respecting New Books : —
Drs, G. E. Eisher and I. J. Schwatt's Textbook of Algebra
with Exercises for Secondary Schools and Colleges . . 251
Relative Motion of the Earth and JEther, by "William
Sutherland 252
NUMBER CCLXXXVI.— MARCH.
Prof. J. J. Thomson on the Theory of the Conduction of
Electricity through Gases by Charged Ions 253
Mr. William Sutherland on Cathode, Lenard, and Rontgen
Rays 269
Dr. R. A. Lehfeldt on Properties of Liquid Mixtures.—
Part III. Partially Miscible Liquids 284
Mr. W. B. Morton on the Propagation of Damped Electrical
Oscillations along Parallel Wires 296
Lord Kelvin on the Application of Sellmeier's Dynamical
Theory to the Dark Lines D,, D 2 produced by Sodium
Vapour " 302
Lord Rayleigh on the Cooling of Air by Radiation and Con
duction, and on the Propagation of Sound 308
Lord Rayleigh on the Conduction of Heat in a Spherical
Mass of Air confined by Walls at a Constant Temperature. 314
Notices respecting New Books : —
Dr. D. A. Murray's Elementary Course in the Integral
Calculus ." 325
Proceedings of the Geological Society : —
Mr. J. E. Marr on a Conglomerate near Melmerby
(Cumberland) , 326
Mr. Beeby Thompson on the Geology of the Great
Central Railway, Rugby to Catesby . . , 327
Prof. T. T. Groom on the Geological Structure of the
Southern Malverns and of the adjacent District to
the West 327
CONTENTS OF VOL. XLV1I. — FIFTH SERIES.
On the Heat produced by Moistening Pulverized Bodies :
New Therm ometrical aDd Calorimetrical Kesearches, by
Tito Martini 329
Combinatiou of an Experiment of Ampere with an Experi
ment of Faraday, by J. J. Taudin Chabot 331
Experiments with the Brush Discharge, by E. W. Marchant . 331
NUMBER CCLXXXVIL— APEIL.
Dr. C. Chree on Longitudinal Vibrations in Solid and
Hollow Cylinders 333
Mr. R. "W. Wood on some Experiments on Artificial Mirages
and Tornadoes. (Plate III.) 349
Mr. J. EoseInnes and Prof. Sydney Young on the Thermal
Properties of Normal Pentane 353
Mr. R. W. Wood on an Application of the Diffraction
Grating to ColourPhotography 368
Dr. G. Johnstone Stoney on Denudation and Deposition. . . . 372
Lord Eayleigh on the Transmission of Light through an
Atmosphere containing Small Particles in Suspension, and
on the Origin of the Blue of the Sky 375
Prof. Oliver Lodge on Opacity 385
Prof. J. J. Thomson : Note on Mr. Sutherland's Paper on the
Cathode Pays 415
Notices respecting New Books : —
Harper's Scientific Memoirs, edited by Dr. J. S. Ames. . 417
Proceedings of the Geological Society : —
Mr. W. Wickbam King on the Permian Conglomerates
of the Lower Severn Basin 417
Dr. Maria M. Ogilvie [Mrs. Gordon] on the Torsion
Structure of the Dolomites 419
NUMBEK CCLXXXVIIL— MAY.
Mr. F. H. Pitcher on the Effects of Temperature and of
Circular Magnetization on Longitudinally Magnetized Iron
Wire 42L
Dr. Edwin H. Barton on the Equivalent Resistance and
Inductance of a Wire to an Oscillatory Discharge 433
Mr. L. N. G. Filon on certain Diffraction Fringes as applied
to Micrometric Observations 441
VI CONTENTS OF VOL. XLVII. FIFTH SERIES.
Page
Prof. Carl Barus on the Absorption of Water in Hot Glass
(Second Paper) 461
Lord Kelvin on the Application of Force within a Limited
Space, required to produce Spherical Solitary Waves, or
Trains of Periodic Waves, of both Species, Equivoluminal
and Irrotational, in an Elastic Solid 480
Dr. C. Chree on Denudation and Deposition 494
Notices respecting New Books : —
Prof. Silas Holman's Matter, Energy, Force, and Work. 496
Messrs. J. Harkness and F. Morley's Introduction to the
Theory of Analytic Functions 497
Proceedings of the Geological Society: —
Mr. H. H. ArnoldBemrose on the Geology of the Ash
bourne and Buxton Branch of the London and North
Western Railway — Ashbourne to Crake Low 498
Prof. J. B. Harrison and Mr. A. J. JukesBrowne on
the Oceanic Deposits of Trinidad 498
A Fivecell Quadrant Electrometer, by Prof. H. Haga .... 499
NUMBER CCLXXXIX.— JUNE.
Mr. Edwin S. Johonnott, Jun., on the Thickness of the Black
Spot in Liquid Films , 501
Mr. Albert Griffiths on the Source of 'Energy in Diffusive
Convection 522
Mr. Albert Griffiths : Study of an Apparatus for the
Determination of the Rate of Diffusion of Solids dissolved
in Liquids 530
Mr. G. A. Shakespear on the Application of an Interference
Method to the Investigation of Young's Modulus for Wires,
and its Relation to Changes of Temperature and Magneti
zation; and a further Application of the same Method to
the Study of the Change in Dimensions of Iron and Steel
Wires by Magnetization 539
Dr. G. Johnstone Stoney on Denudation and Deposition. —
Part II 557
Mr. Gerald Stoney on the Quantity of Oxygen in the Atmo
sphere, compared with that in the Earth's Crust 565
Lord Rayleigh on the Calculation of the Frequency of Vibra
tion of a System in its Gravest Mode, with an example
from Hydrodynamics 566
Notices respecting New Books : —
Dr. F. A. Tarleton's Introduction to the Mathematical
Theory of Attraction 572
CONTENTS OF VOL. XLVII. FIFTH SERIES. Vll
Page
H. Poin care's Theorie du Potentiel Newtonien 573
Dr. Elorian Cajori's History of Physics in its Ele
mentary Branches, including the Evolution of Physical
Laboratories 575
Proceedings of the Geological Society : —
Mr. Prank Eutley on a small Section of Felsitic Lavas
and Tuffs near Conway (North. Wales) 575
Mr. Joseph Thomson on the Geology of Southern
Morocco and the Atlas Mountains 576
Index 577
PLATES.
T. Illustrative of Dr. E. H. Cook's Paper on Experiments with the Brush
Discharge.
II. Illustrative of Dr. van Pijckevorsel's Paper on the Analogy of some
Irregularities in the Yearly Range of Meteorological and Mag
netic Phenomena.
III. Illustrative of Mr. R. W. Wood's Paper on some Experiments on
Artificial Mirages and Tornadoes.
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.]
JANUARY 1899.
I. The Magnetic Fluxes in Meters and other Electrical
Instruments, By Albert Campbell, B.A*
IN all electrical measuring instruments in which the de
flecting or controlling forces are electromagnetic, the
magnetic fluxes and fields are of great importance, and yet
there seem to be no tables published which give even rough
measurements of these, the result being that many people
who are thoroughly expert in the use of instruments have
no idea whatever of the order of magnitude of the magnetic
fluxes occurring in the very commonest instruments. In
order, therefore, to fill this gap to some extent, I have
recently carried out a series of experiments on the subject,
and although the list of instruments thus tested is not very
extensive or complete, I have been able to include in it a
good many of the more familiar types. As individual instru
ments of the same type vary somewhat amongst themselves,
it would have been waste of time to have aimed at great
accuracy in these measurements. Accordingly, whilst guard
ing against large errors in general, I have been content in
one or two cases with results which only indicate the order
of magnitude of the quantity measured.
In most cases the quantity determined has been B, the
magnetic flux density or number of inductiontubes per square
centimetre sometimes through iron, sometimes through air.
* Communicated by the Physical Society : read Oct. 28, 1898.
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. B
2 Mr. A. Campbell on the Magnetic Fluxes in
In some cases <I>, the total flux, was measured, and for several
of the meters determinations of the power lost in their various
parts were also made.
Methods of measuring B.
Method I. Except when the fluxes were alternating, the
method used was the wellknown way with a ballistic galvano
meter. A searchcoil in circuit with the galvanometer either
had the flux suddenly reversed in it, or was pulled quickly
out of the field. For many of the experiments small and
very thin searchcoils had to be used. For instance, in one
of the meters, the available airgap ayrs only 1 millim. across.
The bobbins of these small coils were made by cementing
together one or more round microscope coverglasses between
larger strips of mica for the ends ; they were wound with
from 10 to 200 turns of silkcovered copper wire of 0"075
millim. diameter. Much thicker wire was used when it was
desirable to have the resistance of the galvanometercircuit
low. The smoothness of the mica cheeks allowed the coils to
be withdrawn from position with the necessary quickness.
Fig. 1.
The galvanometer was calibrated from time to time by means
of a standard pair of coils whose mutual inductance was
accurately known. As the measurement of small fluxes by
the ballistic method presents no difficulty, and requires only
ordinary instruments, further description is needless.
Method II. When, however, the small flux is an alter
nating one, the voltage set up in the searchcoil is more
difficult to measure; accordingly two special methods were
here used. In the more accurate of these two methods
the searchcoil Q (fig. 1), through which the flux was made
to pass, instead of being in circuit with a ballistic galvano
meter, was joined directly to a resistancecoil R, laid along
one set of junctions of a minute^thermopile Th, which last
was connected w T ith a sensitive galvanometer G. The re
sistanceware was of manganin (silkcovered) and was placed
along the junctions so as to be noninductive, and to avoid
Meters and other Electrical Instruments. 3
producing eddycurrents. The thermopile consisttd of ten
pairs of thin iron and nickel wires each 7 millim. long.
These metals were chosen as their thermoelectric lines are
far apart, and almost parallel to one another.
Some years ago the writer showed (Proc. Roy. Soc. Edin.
July 1887) that a thermopile used thus could give a fairly
accurate measurement of the current through the resistance
wire, the ultimate deflexion being proportional to the square
of this current. Hence, for a given frequency, the mean
square of the P.D. at the terminals of the searchcoil was
proportional to the deflexion. Each time it was used the
combination was calibrated in the following manner : — A
measured current of 1 ampere from the alternating supply
circuit used was passed through a noninductive resistance of
0*2 ohm, and the resulting P.D. of 0*2 volt was applied to
the ends of the resistance R. From the observed deflexion
of the galvanometer the mean square of the volts per division
was found*. In all cases when the searchcoil was in circuit
the frequency n was observed, being measured by a frequency
teller. If the resistance of the searchcoil be negligible, and
if the flux follows the sine law, we have
r 2 =:10 8 x27r^]Sr 2 B^,
where
v 2 = voltage shown by galvanometer,
n = frequency,
N 2 = number of turns in search coil,
s 2 = area of searchcoil,
and
B = Vuieaii square B.
The other method, which was by means, of a telephone,
was a rougher way, and will be described below.
In Table I. are given some of the results obtained in the
case of the simpler instruments, the third column giving the
resistance of the instrument, the fourth column its full load,
and the fifth the mean value of B at that load. As far as
possible the positions from which the mean value of B were
obtained were chosen so as to give an idea of the working
fluxdensity, and except where otherwise stated the values
given refer to full load.
* The thermopile method of measuring' small voltages is now in lise
in the German Reichsanstalt.
B2
Mr. A. Campbell on the Magnetic Fluxes in
Table I.
No.
Name.
Resistance,
Ohms.
Full
Load.
B (mean).
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Siemens Electrodynamometer
Kelvin AmpereBalance
053
00156
058
00078
324
74
481
000013
30
7X10 5
56
586
13,000
(500)
126
106
10
4 amps.
20 amps.
10 amps.
50 amps.
02 volt
3 volts
100 amps.
18 volts
500 amps.
15 volts
40 volts
Defl.=45°
(100 volts)
(200 volts)
(0  2 amp.)
(1 amp.)
(1 amp.)
80
18
65
55
450
870
400
700
14,200
580
75
70
026
0008
(about 02)
(about 10)
280
3000
20
46
Bifilar Mirror Wattmeter (after ]
Dr. Fleming). J
Ayrton and Mather D'Arsonval 1
Galvanometer. J
Weston Voltmeter
Davies Voltmeter (Muirhead) ...
Ayrton and Perry Magnifying!
Spring Voltmeter. j
Richard Recording Ammeter
Nalder Voltmeter
Any Tangent Galvanometer
Kelvin Astatic Mirror Gralvano \
meter. J
Evershed Ohmmeter, Old Type . . .
,, ,, 'New Type...
Campbell FrequencyTeller
Bell Telephone (double pole)
Ayrton and Perry Variable In ]
ductance Standard. J
Standard Inductance Coil "1
(L= 02 henry. J
As the numbers in the above Table throw an interesting
light on the behaviour of many of the instruments it seems
desirable to discuss them more fully in order.
(1) Siemens Electrodynamometer. — Measurements taken at
the middle and the top of the swinging coil (of 4 turns) in
direction perpendicular to the plane of the fixed coils gave
B = 120 and 40 respectively for the thin coil, and B=21 and
16 for the thick. It will be seen that when the thick coil is
used the deflecting field is quite comparable with the earth's
field.
This, of course, introduces an error with direct currents
unless care is taken to place the instrument so that the
direction of the earth's field is at right angles to that of the
deflecting field, and in the proper sense, i. e. with the instru
ment looking east or west according to the direction of the
current in the swinging coil. The above results show that
the maximum variation at 15 amps, introduced by wrono
placing (viz., due to a field equal to twice that of the earth,
or 0*36) would be about 2*5 per cent, of the mean deflecting
field. This was verified by placing the electrodynamometer on
a welllevelled turntable, and connecting its thick coil with a
Meters and other Electrical Instruments. 5
quite steady source of continuous current by means of twisted
flexible leads. In Table II. are shown the values of the
current indicated by the instrument when turned into various
positions, the first column giving the direction towards which
the front was turned in each case.
Table 11.
Direction.
Apparent
Amperes.
Variation from
Minimum.
K . .
. . 1500
W. . .
, . 1520
13%
s. . .
, . 1532
21%
E. . .
. 1520
13%
It will be seen that the extreme variation is 2*1 per cent.,
w T hich agrees (within the limits of error of the instrument)
with the 2*5 per cent, variation deduced from the observed
fields. For lower currents the variation is much more, being
in the inverse ratio of the current. With the thin coil the
field due to the fixed coil is so much stronger that the varia
tion is slight except at the very lowest currents.
The thick coil had 7 turns and an area of about 46 sq.
centim., and the total maximum flux at full load was found
to be 1850, giving mean B==40. At full load the power
wasted is 6 '2 w r atts and 9*3 watts for the thick and thin coils
respectively.
(2) Kelvin Balance. — The total flux through the central
space of the coils was got by winding the searchcoil round
the supporting pillar, and taking throws by reversing the
current. The resultant flux was about 1600 (for full load).
By the astatic arrangement of the swinging coils the instru
ment is made independent of the earth's field. The self
inductance is about 0'0016 henry.
(3) Bifilar Mirror Wattmeter. — This instrument has ranges
up to 50 or 100 amperes at 10 volts and upwards. The
numbers given refer to the fixed series coil. With direct
currents it is clear that precautions have to be taken to elimi
nate the effects of the earth's field.
(4) D'Arsonval Galvanometer. — This was a ballistic one
(made by Paul) with a narrow swinging coil of the Ayrton
Mather type. The B given in the Table is that in the air
gap in the neighbourhood of the moving coil ; it would seem
to be sufficiently great to be practically unaffected by the
magnetism of the earth. Besides, as it is an instrument for
use in a fixed position, it is only the effect of variable ex
ternal fields that need to be taken into account. This point
6 Mr. A. Campbell on the Magnetic Flaxes in
will be discussed a little further on. To get an idea of how
the flux in the steel varies from point to point along the
annular magnet, an experiment was made with a ringmagnet
of rectangular section, having an airgap as shown in fig. 2.
A small searchcoil which could only just slip along the
magnet was moved by jerks into successive positions, and the
corresponding changes in the flux were calculated from the
throws on a ballistic galvanometer in circuit with the coil.
Fig. 2 shows the result, the radial breadth of the shaded
part being drawn proportional to the flux in the steel at each
position. Fig. 3 is a similar diagram for an ordinary bar
magnet. In the ringmagnet the available airgap flux was
less than one third of the maximum flux at a a.
(5) Weston Voltmeter. — It will be noticed that B in the
airgap of this instrument is very high, viz. 870. This might
lead one to suppose that the earth's field would have no
perceptible effect on its readings, but it must be remembered
that the flux induced in a piece of iron or steel in the earth's
field is usually very many times greater than the flux in air
due to the earth alone. This can be easily shown by connect
ing a coil with a ballistic galvanometer and reversing the
coil with regard to the earth's field first by itself and then
with a soft iron core in it. The throws of the galvanometer
will be enormously increased by the presence of the core.
To find how far the earth's field affected the flux in a perma
nent magnet with a moderate airgap, a coil was wound upon
the circular one shown in fig. 2, and was connected with a
galvanometer. The magnet was then turned round so as to
quickly reverse the action of the earth's horizontal field upon
it. The resulting throw on the galvanometer showed that the
maximum B in the steel, which was about 5000, was only
changed by 3 lines per sq. cm., i. e. by less than 0*1 per cent.
The behaviour of the Weston magnet tallies with this, for when
the instrument, with a steady voltage on its terminals, was
turned round to face each point of the compass, no change
in the reading could be detected, although the scale could be
read to about 1 in 1000*.
(6) Davies Voltmeter. — In this instrument one side of the
rectangular moving coil moves in a narrow cylindrical air
gap between specially shaped polepieces of a strong perma
nent magnet. It gives a maximum angular deflexion of
about 210°.
(7) Ever shed Ammeter. — A coil of 6 turns magnetizes
two small pieces of iron with a movable piece between them,
* At Professor Ayrton's suggestion I have retested the instrument at
the higher readings, and have detected a variation of about Ol per cent.
Meters and other Electrical Instruments. 7
B was measured at about the end of one of the fixed pieces
where the movable piece faces it.
(8) Ayrton and Perry Magnifying Spring Voltmeter. — The
small iron tube, which is surrounded by a coil and pulled
down by it, is 7*2 centim. long, and the iron has a section of
about 0*12 sq. centim. The number given in the table is
the average B in the iron for the whole length of the tube
for a load of 15 volts, i. e. 08 of fall load.
Fiff. 2.
(9) Richard Recording Ammeter. — This has two solid iron
cores (each 4 sq. centims. cross section) round which the
current is carried by a single turn of copper strip ; an iron
armature carrying the pointer is attracted by these cores. B
was measured between one core and the armature. It will
be seen that the total flux is large ; thus a strong deflecting
force is obtained which makes the friction of the recording
pen of less account, but on the other hand much error from
hvsteresis comes in.
8 Mr. A. Campbell on the Magnetic Fluxes in
(10) Dolivo Voltmeter. — Here a thin wire of soft iron is
drawn down into a solenoid. The value given is for the hollow
core of the coil (including the wire).
(11) Nalder Voltmeter. — In this a small piece of soft iron
moves in the magnetic field produced by a coil outside it.
The number in the table is for the middle of the space within
the coil.
(12). Tangent Galvanometer. — When the deflexion =45°,
the resultant field = earth's field = cos 45°
=0*26 (in London).
(13) Kelvin Astatic Mirror Galvanometer. — The galvano
meter was made very sensitive and almost unstable by means
of the controlling magnet. The mean control field (for this
condition) was found by measuring its sensitivity and com
paring it with that when the earth's field alone was used.
The deflecting field for 1° would be less than 0*00002, and
depends on the degree of astaticism of the suspended magnets.
(11) Ever shed Ohmmeter — (Old type; polarized, astatic).
The number given is a rough approximation to the B due to
the shuntcoil alone at 100 volts. About the middle of the
scale the field due to the seriescoil would have the same value.
(15) Ever shed Ohmmeter — (New type, with soft iron
needle). The B given is that due to the shuntcoil alone (at
200 volts). It is clear that the earth's field cannot introduce
much error. In any case the errors due to external fields
can be eliminated (as the makers direct) by reversing the
current and taking the mean of the readings.
(16) Campbell FrequencyTeller. — The B is measured
between the vibrating strip and the attracting pole of the
electromagnet. It will be seen that only a quite moderate
field is required to throw the spring into strong vibration
w r hen it is adjusted to the right length for resonance.
(17) Double Pole Bell Telephone. — The diaphragm was so
close to the poles of the permanent magnet as to form an
almost closed magnetic circuit. A small searchcoil was
wound round one of the polepieces (area = 0*24 sq. centim.),
and the diaphragm was then laid in its place. A throw of the
galvanometer was got by pulling off both the diaphragm and
the searchcoil. Thus the fluxdensity given refers to the
polepieces.
(18) Ay Hon and Perry Variable Standard of SelfInduc
tance. — With the pointer at 0*038 henry, the total flux within
the inner wooden bobbin was about 5200 for 1 ampere.
(19) SelfInductance Standard (L = 0'20 henry).— This
was a coil of 1158 turns of insulated copper wire (diameter
1*24 inillim.), the outer diameter of the coil being 22 centim.
Meters and other Electrical Instruments, 9
and the height 9 centim. The B given in the table is the
average for the whole crosssection of the coil at the middle
of its height. The total flux corresponding to this was 17,500.
B at the centre of this crosssection was 53.
In addition to the instruments already discussed, experi
ments were also made upon a number of meters of different
types, some being for direct, and some for alternating
currents. As some of the types vary in construction from
year to year, a few words of description in each case will make
the results clearer.
Aron Watthour Meter (1894 type). — Range to 50 amps.
at 100 volts. Two pendulums, each carrying shuntcoils,
are acted on by seriescoils under them, one being accelerated
and the other retarded. With both series and shuntcurrents
passing (at full load) the mean B between the fixed and
movable coils was about 70.
Frager Watthour Meter. — Range to 10 amps, at 100 volts.
A meter of the " Feeler " type, in which the shunt and series
coils form an ordinary wattmeter, whose deflexions are
integrated at intervals, The mean B was measured as near
the centre of the shuntcoil as possible.
At full load B = 63,
With shuntcurrent alone . B = 13,
With seriescurrent alone . . B = 50.
Hookham Direct Current Amperehour Meter. — Bange to
100 amps. A small disk, surrounded by mercury which
carries the current, is cut twice by part of the magnetic flux
from a strong permanent magnet (of crosssection 7*5 sq.
centim.). The disk is thus caused to turn. On the same
spindle is a brake disk of copper (5*4 centim. diameter),
which is also cut by a part of the flux from the same perma
nent magnet. Unfortunately it was not possible to take the
meter to pieces, so the driving flux could not be measured.
By slipping a searchcoil along the permanent magnet, the
total leakage was found to be over 26,000 lines. The total
brake flux passes through the disk at four airgaps, two and
two in series, and has a value of about 5000 lines. The
8 polepieces which direct the flux have crosssections of
1*53 centim. each. In two of them the iron near the airgap
is turned down so as to leave only a thin neck of about 0*12
sq. centim. crosssection. This is supposed to increase the
permanence of the flux. At these necks more than J of
the total flux leaks from the iron. Whether they increase or
diminish the permanence seems quite uncertain. The mean
flux density at the four airgaps was found to be about 1020.
10 Mr. A. Campbell on the Magnetic Fluxes in
The power spent in the meter at full load = 12*8 watts.
The power spent in turning the spindle (at 2'2 revolutions
per second) with full load was measured by the method
(1) described below, and was found to be 0*016 watts*.
Hence the efficiency of the meter as a motor = 0*125 per cent.
Kelvin Amperehour Meter. — "Range to 600 amperes. In
this a thin iron core, kept highly magnetized by a shunt
current, is drawn down into a solenoid which carries the main
current. The solenoid had 6 turns (i. e. 3600 ampereturns
at full load), and was about 16 centim. long. At full load
the flux density at the lower end of the solenoid was over
250. With the shuntcurrent alone the total flux through the
moving coil and core was about 330.
Eliliu Thomson Watthour Meter. — Range to 50 amperes
at 100 volts. This meter consists of a small ironiess motor, in
which the seriescurrent goes through the field magnet coils
and the shuntcurrent through the armature. On the arma
ture spindle is a brake disk of copper (13*5 centim. diameter),
wbich passes between the narrow airgaps of three permanent
magnets of the shape shown in fig. 4. These magnets are
often of different strengths, being chosen to give the proper
brakeforce for each individual meter. Their polar faces are
about 7*5 sq. centim. The mean B in the airgaps was about
700. By the method of placing the magnets the greater part
of the flux acts on the brake disk. By slipping a searchcoil
along one of the magnets it was found that the total flux at
a was about 15000, making B about 7000. Of this flux
nearly one half remains in the steel as far as the section at b.
Fig. 4.
Driving Flux. — Without the shuntcurrent the full load
current gives mean B == 130 along the axis of the seriescoils.
The shuntcurrent at 100 volts gives a field at right angles
1o this in which B = 10. The shuntcurrent also passes
through a " compounding " coil fixed coaxially in one of the
* After the author had measured the motorefficiencies of several
meters, M?\ Sidney Evershed somewhat anticipated him by announcing
(Institution of Electrical Engineers, May 12th, 1898) one or two similar
results, making no mention, however, of the method by which the results
were obtained.
Meters and other Electrical Instruments.
11
seriescoils for the purpose of overcoming friction at the lower
loads. The B due to this starting coil alone is about 3.
There is a small stray field from the seriescoils perceptible at
the brake disk, but as it is less than the ^Jq part of the field
due to the permanent magnets its effect may be neglected.
Effect of the Earth's Field. — Since the driving flux density
at full load is only 130, it is clear that at the lower loads the
rate of the meter may be considerably affected by the earth's
field. To test this point the meter was levelled up on a turn
table (as in the case of the electrodynamometer already
described), and a constant load of 4*625 amperes at 100 volts
was kept on it. The load was measured by a Kelvin balance
and a reflecting multicellular voltmeter. The rate of the
meter, i. e. spindle revolutions per wattsecond, was then
determined
(A) with alternating current :
(B) with direct current, the earth's field helping the driving
field ;
(C'i the same, with the earth's field opposing the diiving
field.
Table I1L shows the results of the tests.
Table III.
Position of Meter.
Current.
Rate.
(A) Facing East.
, (B) „ West.
(C) „ East,
Alternating at 80^
per second.
Direct.
Direct,
0C002125
00002135
00002212
It will be seen from (B) and (C) that by turning the meter
round through 180° its rate at ^ l° a ^ ^ s altered by 3*6 per
cent., which is exactly what might have been predicted from
the value of B given above. The rate with alternating current
does not lie between the rates (B) and (C) as it ought, but is
about 5 per cent, slower than (B).
Power Spent in Meter. — The resistance of the seriescoil
= 0*0066 ohm, and that of the shuntcoil = 2030 ohms ;
hence the power spent in heating the coils =16*5 + 49 = 21*1
watts.
The power spent in actually driving the meter (at full load)
was measured by two methods as follows : —
(1) An arm of about 10 centim. long was attached to the
spindle of the meter (at right angles to it). With full load
12 Mr. A. Campbell on the Magnetic Fluxes in
switched on, the tangential force /necessary to hold at rest
the end of this arm was measured by the extension of a spiral
spring which had been calibrated by known weights.
Then
n
Power (in watts) = 10 7 ./. 27rr . — ,
where
r = length of arm,
 = number of revolutions per second when the spindle
is free to move.
(2) In the second method the shuntcircuit was disconnected
and joined directly to a sensitive galvanometer with a resist
ance of 12,000 ohms in circuit. A measured current was
sent through the seriescoils, and the spindle was turned at
about the full rate. Thus the meter acted as a dynamo and
gave a deflexion on the galvanometer. By watching the gal
vanometerscale it was not hard to keep this deflexion steady,
and from the known calibration of the galvanometer the
voltage given by the armature was obtained. The total
E.M.F. multiplied by the normal shuntcurrent (0*0493 amp.)
gave the drivingpower.
Method (1) gave for the drivingpower at full load 0*020
watt, whilst method (2) gave 0*021 watt. Hence the efficiency
of the meter as a motor is only 0*095 per cent
Hoohham Alternating Current Watthour Meter. — Eange to
10 amperes at 100 volts. In this meter a solid iron core
forming an almost closed magnetic circuit is magnetized by
a shuntcoil, which latter, by reason of its large inductance,
carries a current which lags behind the main current by
50° to 60°. A smaller Ushaped piece magnetized by the
main current has its poles close to the upper pole of the
larger iron core, and one of the poles carries a copper screen.
A small aluminium disk (8*8 centim. in diameter), partly
between the poles, is turned by the rotary magnetic flux
thus produced. The brake force acts on the same disk, and
is due to a tall permanent magnet about 20 centiin. high
(with a narrow airgap) , as shown in fig. 5.
Brake Flux. — The maximum flux in this magnet was
found to be near P, and had the value 19,820, corresponding
to B = 8800. Of this only 4400 lines ultimately cut the
disk, much being lost by cross leakage from Q and S to the
opposite limb. Thus less than J of the maximum flux is
made use of. The mean B between the poles was found to
be 650.
and other Electrical Instruments.
13
Driving Flux. — The distribution of the somewhat compli
cated alternating field was traced by the following method,
which also gave rough quantitative results. A telephone T
Fi<r. 5.
*±^r\
l&AO
S
(fig. 6) was arranged in a circuit with the searchcoil F and
the lowresistance strip H K in such a way that the strip
could be switched out of circuit at will. One of the con
nexions to HF was through a sliding contact, so that the
Fiff. 6.
resistance of the part in the telephonecircuit could be varied
from 0*05 ohm downwards. A current of 1 ampere was
maintained in the strip and was from the alternating circuit
which supplied the meter load. With the strip out of circuit
the searchcoil was moved into various positions, and the dis
tribution of the alternating flux observed by means of the
sound in the telephone. To measure the flux at any position
H K was set to such a value that the small P.D. introduced
by it into the telephonecircuit gave a sound of the same
loudness as that given when the searchcoil was placed in the
flux. The absolute values given by this method came out 10
14 Ml*. A. Campbell, on the Magnetic Flaxes in
to 15 per cent, too small ; but it proved a very convenient way
of comparing the fluxdensities at various positions. It was
thus found that when the main current (full load) was switched
on in additioLi to the shuntcurrent, the fluxdensity under
the middle pole was nearly doubled. The most curious fact
brought to light by the method was that across the airgap of
the permanent magnet a considerable alternating flux is in
duced, the value of B at that position being actually about
5 of that between the poles of the shuntmagnet with the
shuntcurrent alone. The permanent magnet forms a kind of
secondary magnetic circuit directing around itself the alter
nating eddycurrents in the disk. Whether this has any
sensible demagnetizing effect upon the permanent magnet the
writer has not determined.
The fhixes were measured more exactly by the method of
the searchcoil and thermopile described above. With shunt
alone the root of mean square B was about 50 in the airgap
and 800 in the iron core just above the shuntbobbin. The
shuntcurrent at 100 volts (86 ~ per sec.) was found to be
0'031 ampere. If the current followed a sine curve its maxi
mum value would be 0*044. When a direct current of this
value was tried the fluxes produced were much larger than those
with the (supposed) equivalent alternating current. This is
partly due to the fact that the shuntcurrent does not follow
the sine law, but is no doubt also due to the existence of eddy
currents in the iron core. That these currents even in the
disk spread the flux anyd reduce the fluxdensity in the airgap
was shown qualitatively by placing a searchcoil, connected
with a telephone, in the airgap of a ring electromagnet
excited by alternating current. The sound in the telephone
was lessened when a copper disk was held near the coil in the
airgap. The searchcoil and thermopile method showed that
the fluxdensity had been reduced by 8 per cent.
To find how the coreflux varied with alteration of the
voltage on the shuntcoil, by the same method the B just
above the shuntbobbin was measured for a number of voltages
from 20 up to 100 volts. It will be seen from the curve in
fig. 7 that B is very neaily proportional to the potential
ditference. In practice it is found that the speed of rotation
is very nearly proportional to the voltage.
Power spent in Meter. — To approximate to the amount of
power spent by reason of hysteresis when the shuntcurrent
alone is on, the iron was carried through a cycle by means of
direct current of such amount as to give a maximum B nearly
corresponding to that given with alternating current at
100 volts. The curve obtained is shown in fig. 8.
Meters and other Electrical Instruments,
U
Fig. 7,
1000
50
Yolts on Shuntcoil.
Fig
.8.
1500
B
1000
/
500
/
/
004 0
03
02 0
Y o
/
/
■01
Cc//?ff£A/T (/,
02
v Shi/a/tJ
03 ' 004
//
16 Mr. A. Campbell on the Magnetic Fluxes in
From this curve it is not possible to get the exact value of
the hysteresis loss, as the iron core is not a uniformly mag
netized closed circuit. To get an idea, however, of the amount
of the power wasted by hysteresis, let us suppose that the
magnetic circuit is equivalent to a uniform iron ring of the
same crosssection as the core of the shantcoil, uniformly
wound with the same number of turns n x as the shuntcoil *
(carrying the same current) , and traversed by a flux equal to
that at the part of the metercore for which the curve in fig. 6
was taken.
Let c = current at any moment,
anci s = section of ring.
Then
Hysteresisloss in joules per cycle
= §cdv = 10"% jc^ = W\s§cdB
= lO" 8 ^ x (area of curve)
= lO" 8 x 1200 X 37 x area
= 000114.
.*. at 86^~ per second,
Power spent = S6x 0*00114 = 0'098 watt.
It was found by direct measurement that the actual power
spent in the meter (with the shuntcurrent alone) was far
larger than this. Accordingly measurements were made of
the rise of temperature of the iron core by means of an iron
nickel thermopile. Three junctions of the pile were bound
against the iron, which had its surface well insulated with
paint. A pad of wadding was tied over the spot, and the
thermopile was connected with a suitable galvanometer. The
shuntcurrent was switched on for 120 seconds, the deflexion
being read at intervals, and on breaking the current the
cooling was observed for several minutes. The thermopile
and galvanometer were calibrated with a known difference of
temperature, and the curves of heating and cooling were
drawn. The curve of heating was then corrected by means
of the other curve, and thus the heating of the iron (cor
rected for cooling) was found.
At a spot just above the shuntbobbin the corrected rise of
temperature w r as o, 87 C, whilst near the airgap it was only
about half of this. Taking account of this last fact, we may
* The exact value of n x was not known, but the value (1200) used
was estimated from the resistance and gauge of the shuntwire and the
size of the coil.
Meters and other Electrical Instruments. 17
take that a volume of about 61 c.c. was raised in temperature
by 0°'87 C.
Now
rower spent = 4*2 watts,
where <r = sp. heat of the iron,
V = volume „ „
D = density „ ,,
^T= temperaturerise (corrected),
t = time in seconds;
therefore from the results above,
Power = 1*58 watts.
Since the hysteresisloss is about 0*10 watt, it will be seen
that the eddycurrent loss = 1*48 watts.
The power lost by eddycurrents in the disk was similarly
measured, and was found to be about 0*02 watt.
In the shuntcoil the C' 2 R loss =0*57 watt ; and hence the
total copper and iron losses =2*17 watts.
Direct Measurement of Total Power. — The total power
given to the meter was measured by the threevoltmeter
method, in which was used a reflecting electrostatic voltmeter
accurate to about 1 in 1000 at all the points of the scale
required. The result obtained was 2*06 watts, which agrees
fairly well with the total watts shown by the other methods.
Driving Power. — The drivingpower at full load was also
measured by the method of the springbalance described
above ; it was found to be 0*00073 watts. Taking account
of the seriescoil (whose resistance was 0*009 ohm) the total
power spent at full load is over 3 watts ; and hence
Motor efficiency = 0*024 per cent.
Scheefer Alternating Current Watthour Meter. — Range to
10 amperes at 200 volts. In this meter a pile of Eshaped
iron stampings has the seriescoil on one outer limb and the
shuntcoil on the other. An aluminium cylinder is turned
by the rotary field thus produced, and on the same spindle is
a brake disk with a single magnet exactly like those in the
Elihu Thomson meter.
The mean B between the poles of this magnet was about
480. With the shuntcurrent alone the v/mean B 2 between
the shuntpole and the driving cylinder was 570. The total
power spent with the shunt alone on is 11*5 watts, of which
7*1 watts are due to C 2 R loss.
Shallenberger Alternating Current Amperehour Meter. — ■
Range to 20 amperes. A seriescoil (of about 50 turns) and
Phil. Mag, S. 5. Vol. 47. No. 284. Jan. 1899. C
18 Magnetic Fluxes in Meters and other Electrical Instruments.
a small shortcircuited secondary coil, with their axes at an
angle of about 45°, produce a rotary field by which is turned
a small disk with a soft iron rim. The brakeforce is obtained
by airfriction on four aluminium vanes.
By the searchcoil and thermopile method it was found that
inside the seriescoil
VmeanB 2 =100.
Power spent in Meter. — The resistance of the seriescoil
was 0'025 ohm ; hence the power spent in it =10*0 watts.
The power spent in the copper stampings which form the
secondary coil was found by measuring their rise of tem
perature with a small copperiron junction. This rise (at
full load) was found to be about o, 33 C. per minute, the
cooling being negligible. The volume of copper was about
21 '4 c.c. ; whence the power spent = 0'40 watt.
The drivingpower was found to be 0*0069 watt ; therefore
the motor efficiency =0'066 per cent.
Current in Secondary Coil. — The current in the short
circuited secondary coil could not be measured directly.
Calculating from the dimensions of the coil, however, the
resistance was foimd to be 8*5 x 10 6 ohm. From this and
the value of the power (0*40 watt) we find that the secondary
current attains the extraordinary value of 220 amperes.
For the sake of comparison some of the above results are
collected in Table IV.
Table IV.
Name.
Driving B.
Brake B.
Power
spent.
Motor
Efficiency.
Elihu Thomson
130
(not
measured)
50
100
700
1020
650
watts.
214
128
32
104
per cent.
0095
0125
0024
0066
Hookham (direct curr.) ...
Hookham (alternat. curr.)
Shallenberger
In conclusion, it will be noticed that in motor meters the
drivingflux density is of the order 100 and the brake B from
500 to 1000 ; also that the motor efficiencies are all very
small, particularly in the case of the alternatingcurrent
meters. In all of them the greater part of the power taken
is spent in heating conductors (either by eddycurrents
or otherwise). If a small fraction of this wasted energy
could be employed to overcome with certainty the friction at
the lowest loads, a great advantage would be gained thereby.
June 7th,
[ 19 ]
II. A Resonance Method of Measuring En^gy dissipated
in Condensers. By Edwakd B. Rosa and Akthur W.
Smith *.
THAT the dielectric of a condenser becomes warmed when
an alternating electromotive force is applied to the
terminals of the condenser has long been known, and the
study of this heating effect has been undertaken by numerous
observers. Kleiner f used a thermoelectric couple imbedded
in the dielectric to measure the rise of temperature, and noted
a considerable heating effect in ebonite, gutta percha, glass,
and mica, but none at all in paraffin and kolophonium. He
reports that in spite of all attempts by variations of the con
ditions of the experiment, no heating could be detected
in these two lastnamed substances. On the other hand,
Boucherot % has made paraffinpaper condensers for use on the
3200volt commercial circuits of Paris, some of which became
so hot in use that they were obliged to be cut out. Boucherot
says of the heating effect that if a condenser rises as much as
30° C, it should be rejected. That this is good advice is
evident from the fact that paraffin melts at 54° C. ; and hence
when 30° C. above the temperature of a summer's day (say,
25° C. or 77° F.), the paper would be floating in melted
paraffin.
Bedell, Ballantyne, and Williamson § report experiments
upon a paraffinpaper condenser of 1*5 microfarad capacity,
the efficiency of which was found to be 95*6 °/ , or 4*4 °/
lost in heat. The loss was determined by a threevoltmeter
method, similarly to measurements on a transformer. It was
put upon a 500volt circuit at a frequency of 160, and the
current was therefore about 07 ampere, the apparent watts
about 350, and the heating effect 15*4 watts. The tempe
rature rose one degree per hour.
Threlfall ] reports a test on a paraffinpaper condenser of
his own construction which had a capacity of 0'123 micro
farad, and on a circuit of 3000 volts and a frequency of 60
the rise of temperature was less than onefifth of a degree per
hour. The apparent watts would be about the same as in the
experiment of Bedell, Ballantyne, and Williamson, although
the capacity was only onetwelfth as much. Threlfall concludes,
apparently, that since his condenser had only onetwelfth the
* Communicated by the Authors.
t Wied. Ann. vol. 1. p. 138.
% LEclairage Electrique, Feb. 12, 1898.
§ ' Physical Review,' October 1893, vol. i. p. 81
 ' Physical Review/ vol. iv. p. 458 (1897).
C2
20 Messrs. E. B. Rosa and A. W. Smith on a Resonance
electric capacity, it also had only one twelfth the capacity
for heat of the other condenser ; and since the temperature
rose five times more slowly, therefore that the percentage loss
of energy was only onesixtieth as great, that is, £$ of 3 %,
or 0*05 °/ , giving his condenser an efficiency of 99*95 °/ !
The reasoning is, however, quite inconclusive, for nothing is
said concerning the thickness of the paraffinpaper dielectric
and the volume of the condenser. Suppose the dielectric of
the smaller condenser to be 0'0129 inch thick, which is three
times the thickness of the other. (It might have been as
thick as that, seeing it sustained for a considerable period an
alternating voltage of 3000.) Then the volume per unit of
capacity would be nine times as great as the other, and the
heat required to raise its temperature as rapidly would be
nine times as great, assuming the capacity for heat and rate
of radiation the same for both. The loss would then have
been 0*45 % instead of 005 %. This illustrates how entirely
inconclusive any determination merely of rise of temperature
is, unless all such circumstances as heat capacity, rate of
cooling, thickness of dielectric (or intensity of the electro
static induction) are specified.
The enormous discrepancy between the results above re
ferred to as to the quantity of the heating effect in condensers,
and the almost entire lack of precise statements as to its
numerical value, led us to undertake, more than a year ago,
to measure this energy loss in such a way that it could be
expressed absolutely.
We proposed to measure by means of a wattmeter the
energy dissipated in a condenser when it is subjected to an
alternating electromotive force. In order that the frequency
of charge and discharge be perfectly definite, the electromotive
force should be a simple harmonic one, that is, the upper har
monics of the fundamental should be absent. This is most
easily effected by inserting in the circuit in series with the
condenser a coil of wire having large selfinduction, but
without an iron core. The variable permeability of the iron
will give rise to upper harmonics, especially if the magnetic
induction of the core attains large values ; and hence a coil
without an iron core is necessary. Moreover, if the self
induction is not large enough it will reinforce some of the
upper harmonics instead of quenching them; and the presence
of the coil will be detrimental rather than beneficial. The
best value of the selfinduction of the coil is such that the
fundamental is reinforced to a maximum degree; in other
words, it is that value of L given by the equation
£=2ttx/LC;
Method of Measuring Energy dissipated in Condensers. 21
where t is the period of the fundamental component of the
impressed electromotive force and the capacity of the
condenser.
There is another practical advantage resulting from this
arrangement, namely, that the resulting resonance raises the
electromotive force at the terminals of the condenser very
greatly, saving the necessity of raising the voltage by trans
formers. And a third advantage now appears in the fact that
the wattmeter may be inserted across the lowvoltage supply
wires to measure the total power expended upon coil and
condenser. Then subtracting the IV loss of the coil, the
remainder will be the power expended upon the condenser.
This supposes, of course, that there is no iron core and no
eddycurrent loss in the copper coil itself.
The Resonance Method,
Fig. 1 shows the connexions for this method with the coil
in series with the condenser. M, N are the lowvoltage
supplywires of an alternating circuit, S is an adjustable re
sistance, E is a dynamometer, F a wattmeter, R a non
inductive resistance in shunt with the coil and condenser,
Fig. 1.
N
Dyn.
R
"W
r *
p ■
lltht
1 B
V V V
\XXUJ
Coil
Condenser
joining the points A and D. The wattmeter therefore mea
sures the power expended between the points A and D,
including the Vr w loss in the fixed coil of the wattmeter, but
not including the i 2 J& loss in the shuntresistance R. The
13U5
•22 Messrs. E. B. Rosa and A. W. Smith on a Resonance
energy expended in the resonance coil and the fixed coil of
the wattmeter, I 2 (r c \r w ), subtracted from the total power
measured leaves the condenser loss. This remainder is found
to be also proportional to the square of the current. Hence
we may write it I 2 r s , and r s is the u equivalent resistance " of
the condenser, which it is desired to find. This does not
indicate the nature of the process by which energy is dissi
pated in a condenser, but simply that for a given condenser
made of a given dielectric at a given temperature and fre
quency, the heating effect is the same as though there were a
certain resistance r 8 in series with a perfect condenser of the
same capacity. For the same dielectric r s changes with
changes in the temperature or the frequency; and for another
dielectric with the same capacity, temperature, and frequency
r s would be different.
If £ = 27r\/LO, there is complete resonance; t= , and
1 . 1 n
p = 27rn. Therefore  = n/LC, or 0=^. That is, for
p p 2 h '
complete resonance the capacity is inversely proportional to
the square of the frequency for a given selfinduction. If
Fig. 2.
(b)
B b
A a
a c D
the frequency is fixed, either the capacity or the induct nee
may be varied until the current is a maximum ; but if the
frequency can be varied, the maximum resonance may be
attained without varying C or L.
Method of Measuring Energy dissipated in Condensers, 23
In fig. 2, let Aa = i*c, the ohmic resistance of the coil;
aB = pL, the reactance of the coil ;
Bb=r s , the equivalent resistance of the con
* denser ;
bG = 7s, the condensance of the condenser ;
pyj
CD = r w , the resistance of the fixed coil of the
wattmeter, its induction being ne
gligible ;
AD = the resultant impedance of the circuit.
Then if pL= p, the reactance is equal and opposite to the
condensance (fig. 2 c) , the resonance is complete, and the
impressed electromotive force e is expended in overcoming
the resultant resistance AD=r c + r s +r. and 1= —   — .
r c + r s + r w
lines represent
In fig. 3, similar to fig. 2 c, the several
electromotive forces. Aa = Ir c = e c = that part
of the electromotive force expended in over
coming the ohmic resistance of the coil ; simi
larly, B# = e s , OD = e w . Ba is the electromo
tive force (due to resonance) which overcomes
the reactance, and bG is the electromotive force
which overcomes the condensance. The po
tential of the point B varies through a wide
range; whereas the points A, C, and D, and
the instruments suffer only small changes of
potential.
Fig. 3.
B
/
Advantages of the Method.
Herein lies one of the chief advantages of
the method, that voltages below a hundred
have to be dealt with at the instruments,
whereas upon the condenser there may be an
active electromotive force of several thousand
volts. The noninductive resistance E, is at
most a few hundred ohms. On the contrary, if the shunt
resistance were applied directly at the terminals of the con
denser, it would necessarily be several thousands of ohms, it
must be capable of carrying the entire shuntcurrent of the
wattmeter, it must be strictly noninductive, and be of
known value — conditions difficult to fulfil. In the resonance
method a small inductance in the shuntresistance or the
movable coil of the wattmeter produces no appreciable error;
whereas in the simple wattmeter method it produces a large
error.
EMF
24 Messrs. E. B. Rosa and A. W. Smith on a Resonance
To illustrate this point let us take a special case. Suppose
the resistance R is 500 ohms and the inductance of the shunt
circuit, including both the resistance R and the movable coil
of the wattmeter, is 0'003 henry, %7rn being 800. Then
^L = 24 and tan (^ = 00048, ^ = 16' 30", the angle of
lag of the shuntcurrent behind the electromotive force.
Suppose the true angle of the condensercurrent, <£ 2 (fig. 4), is
89° 40' ahead of the electro
motive force. Then the dif
ference of phase of the two
currents in the wattmeter will
be = ^ + 03 = 89° 56' 30",
and the power factor, cos cf>,
of the expression watts =
EI cos^will be cos 89°56 / 30"
instead of cos 89° 40', that is
•00102 instead of '00582; thus
the wattmeter would indicate
that the power absorbed in the
condenser was only about one
sixth of what it really is. If
the lag of the shuntcurrent
were more than 20 x (a not
improbable value in many
cases), the deflexion of the wattmeter would be negative !
It is possible that this explains why it has often been claimed
that the loss in certain condensers is too small to be measured
by a wattmeter. For example, Swinburne*, speaking of
some of his own condensers, says " a condenser that takes
2000 volts and 10 amperes has a loss that is too low to measure
— that is to say, it is less than 5 or 10 watts."
On the other hand, if by means of a resonancecoil the
current and electromotive force have been brought very
nearly, if not exactly, into phase, any small lag of the shunt
current will make no appreciable error. Thus, the cosine of
10° is 9848, and of 10° 16' 30" is '9840, a difference of less
than one part in a thousand.
In order to determine the precise values of r e and r w a
Wheatstone bridge is joined to A D and the condenser short
circuited, so that the resonancecoil and the fixed coil of the
wattmeter and the leadwires form the fourth arm of the
bridge. The resistance is then quickly measured just after
the wattmeter has been read and the alternating circuit broken,
and changes due to temperature are included.
* < Electrician/ Jan. 1 (1892).
Method of Measuring Energy dissipated in Condensers.
Difficulty of the Method.
We have seen that the presence of a resonancecoil in series
with the condenser (1) quenches, to a large extent at least, the
upper harmonics, (2) raises the voltage upon the condenser,
thus avoiding transforming up, (3) enables measurements to
be made more safely and more conveniently upon a low voltage,
and (4) transfers the wattmeter problem from the most un
favourable case (where the angle of phasedifference is nearly
90°) to the most favourable case where the current and
electromotive force are nearly in phase. There is, however,
one serious difficulty in the method. If the resonance coil is
made of small wire, it has a great resistance, and of the total
power measured only a small part is expended on the con
denser. Thus the condenser loss is the difference between
two relatively large quantities, and cannot be determined as
accurately as would be desired. If, on the other hand, a
large coil of larger wire is used so that its resistance is small,
there will be eddycurrents in the copper of the coil, and the
power expended on the coil will be greater than 1V C . This
excess will go into the remainder as condenser loss, and may
give rise to a considerable error. If the wire is of large cross
section, but stranded, so that its resistance is small and the
eddycurrents negligible, then a large coil will have a large
inductance, and no difficulty appears. The method is then
accurate as well as quick and convenient.
The Resonance Ratio,
As the condenser is alternately charged and discharged
energy is handed to and fro between the coil and the condenser.
When the condenser is charged to its maximum extent the
current is zero and all the energy is potential and residing in
the condenser. A quarter of a period later the condenser is
discharged and the current is a maximum ; the energy is now
kinetic, and resides in the magnetic field of the resonance
coil. At other instants the energy is partly potential (in the
condenser) and partly kinetic (in the coil). As this transfer
of energy to and fro continues, the dynamo supplying the
current furnishes just enough energy to make good the losses,
that is, the heating effect in the wires and the dielectric of the
condenser. The losses due to electromagnetic radiation and
mechanical vibrations are usually negligible.
For the condenser alone,
i= E
\A + <y
cy
26 Messrs. E. B. Rosa and A. W. Smith on a Resonance
I and E being the square root of the mean square values as
indicated by an electrodynamometer and electrometer.
For the combined circuit
1 =
\/(r w + r e + r s y+(p^^j
where e is the small impressed electromotive force and the
denominator is the combined impedance of the circuit.
For complete resonance,
jt)L= p, and hence 1=
Cp' r w + r e + r a
Hence
/TTj~
E V c C 2 p 2 Impedance of the condenser ^
— = — = —  — rfn—i rr =±tesonance ratio.
e r c + r s + r w 1 otal resistance
In one case e was 50 volts and E was 2250, giving a reso
nance ratio of 45. The impedance was 51 ohms, r +r w
51
was *38, r s was '72. Hence ' =46*4, agreeing very
nearly with the ratio of the voltages. In this case the coil
was of large wire (No. 5 B & S) , and had considerable eddy
current loss. Hence the value '72 for r s was too large, and
the degree of resonance was lower than it would have been in
the absence of eddycurrents. In another case, using a coil of
No. 10 wire, the impressed electromotive force was 29*5 volts,
the voltage on the coil or the condenser was 1808, and the
E
resonance ratio — was therefore 61*9.
e
The Resonance Coll in Parallel.
A second arrangement of the resonancecoil is to put it in
parallel with the condenser (fig. 5), and impress upon both a
high electromotive force. Each of the two parallel circuits
from B to C takes its own current, independently of the
other, but being nearly opposite in phase they nearly cancel
each other in the supply wires. Hence a small transformer
is sufficient to supply the small current needed, although
without the resonancecoil a large transformer would be
necessary. If, as before,
pL= k'
Method of Measuring Energy dissipated in Condensers. 27
there is complete resonance. The two parallel circuits having
the same impedance take the same current ; one current is
nearly 90° ahead in phase of the electromotive force (see
Fig. 5.
N
WM
Condenser
Dyn.
Coi?
curve 2, fig. 6), the other is nearly 90° behind (curve 3), the
sum of the two being relatively very small (curve 4) and in
phase with the impressed electromotive force (curve 1). The
shunt resistance must be applied at the high voltage terminals,
but as a small amount of selfinduction produces no appre
ciable error in the wattmeter, the movable coil may be long
enough to make the wattmeter quite sensitive, and so a quite
small shunt current may be used. This requires a larger
resistance, but with much smaller carrying capacity, since a
much smaller shunt current will suffice than when the main
current differs largely in phase from the shunt current.
To illustrate this point, suppose as before that for a given
condenser the angle of advance of the current is 89° 40'.
The power factor, cos <j>, in the expression watts = E I cos <£
is in this case '00582. If now a resonancecoil be placed in
parallel with the condenser and the current in the fixed coil
of the wattmeter brought into phase with the electromotive
force, then cos <£ = 1. To get a certain deflexion of the watt
meter, therefore, we must have the product of the two currents
in the wattmeter nearly 200 times as great in the first case
28 Messrs. E. B. Rosa and A. W. Smith on a Resonance
Fig. 6.
Method of Measuring Energy dissipated in Condensers. 29
as in the second, and this requires a relatively large shunt
current.
A modification of the method, if a second small transformer
is available, is to transform down again to a low voltage, and
put the shunt circuit of the wattmeter on the low voltage
secondary of this second transformer. The current will now
be almost exactly opposite in phase to the high electromotive
force at the terminals of the condenser, and by interchanging
the terminals the wattmeter deflexion will be the same as
before, if the shunt resistance is reduced in the ratio of trans
formation. The currents in the two coils of the wattmeter
are so nearly in phase with one another that a small change
in the phase of the shunt current will produce no appreciable
error.
The Efficiency of a Condenser.
Having thus determined the energy, w, dissipated in a
condenser, by wattmeter measurements, we
readily find r s , the equivalent resistance of the
condenser, from the expression
10= IV.
The ratio of the equivalent resistance to
c P
is cot $ (fig. 7) , <f> being the angle of advance
of the current ahead of the electromotive force.
It remains to calculate the efficiency of a con
denser.
In fig. 8 I is the current flowing into and
out of the condenser, assuming both current
and electromotive force to be simply harmonic.
The dotted curve is the power curve.
^=E X sin pt j
where 0=the instantaneous E.M.F. acting on
the condenser, and E x is its maximum value.
E x
sin (pt{cj)) = I 1 sin {pt + <f>),
Fig. 7.
Impedance
£?' =E 1 I 1 sin pt sin (pt + </>)
= EJi [sin 2 pt cos $ + sin pt cos pt sin <£] .
The area of the power curve for one halfperiod, that is the
area of the positive loop from B to C, minus the area of the
30 Messrs. E. B. Rosa and A. W. Smith on a Resonance
Fig. 8.
■><?^.
Jfethod of Measuring Energy dissipated in Condensers. 31
negative loop from C to D, is given by the integral
I ^'^=E 1 I 1 cos^>l sin 2 p£ ^ + E^ sin </> sinptcosptdt,
EJi jFpt sin2»rr EJj . ,r _ 1»
1 EJj ,
and this is the work done in — of a second.
Hence for one second the work done, and therefore the
power, is the wellknown expression
P^Ii cos </> = EI cos (E = Ei t/I),
E and I being the effective values of the electromotive force
and current.
The area of the positive loop, which represents the work
done upon the condenser in charging it, is
and the area of the negative loop, which represents the work
done by the condenser in its discharge, is the value of the
same integral between the limits tt— <£, and ir, for pt.
The efficiency is the ratio of the work done by the condenser in
its discharge to the work done upon the condenser in charging
it.
This is the gross efficiency, r\.
mi . oqo Area of the Negative loop _ l/^E 1 I 1 x "48638 _ Q , „
Taking <£ = 89°, V = Area f Positive loop " l/pE&x '51378 ~ 94 6 ? %
i. *=87°,? = „ = S= 84 * 75 %
„ (£ = 45°, *? = = = 638 o/
Having found the angle <£ by the wattmeter measurements
the gross efficiency of the condenser may be found from the
above equations, or taken from fig. 9, which is drawn from
them.
32 Messrs. E. B. Rosa and A. W. Smith on a Resonance
Fig. 9.
90
SO
60
50
V
\
\
\ \ *
^
%
\ n
\ *
\ c>
\ %
Sl
\ <*
\Sf
\
\
\ v
\
\
85° 80°
7o z
70 c
65° 60° 55°
Angle (p.
50°
45 c
40°
35° 30 c
Second Definition of Efficiency. — Net Efficiency.
Regarding the condenser as an instrument for storing
electrical energy, and one in which a certain amount of
energy is dissipated in the process, we may define the per
centage of loss as the ratio of the energy dissipated to the
energy stored. The efficiency is then unity minus the loss,
or, in per cent., the efficiency is 100 minus the per cent, of
energy dissipated. This we may call the net efficiency, and
represent it by e. Then
Energy Stored — Energy Dissipated 1 Energy Dissipated
Energy Stored ; Energy Stored
In fig. 10,
I 1 r g = E 1 cos <£ = active electromotive force ;
^=E 1 sin <£ = wattless electromotive force, or the E.M.F.
which charges the condenser.
The maximum charge of the condenser is
CE X sin <£,
6 =
Method of Measuring Energy dissipated in Condensers. 33
and its energy Fig. 10.
W=iOB 1 2 sin 2 0. r rs
The expression for the power is
JliEj cos (j>,
But , ^^ . ,
I 1 = CE 1 sm</>.p.
Hence the energy dissipated per second is
•JCEi 2 sin (j> cos (j> .p,
and the energy dissipated per halfperiod (that
is during the time of a single charge and dis
charge) is
zy = iCE 1 2 sin <j) cos cj> . it.
The relative loss is therefore
^ = 7TCOt(/>,
and the net efficiency is
6=1 — TTCOt (f>.
For
= 89°, 6 = 9452%,
= 88°, e=8903 / ,
<£ = 87°, 6 = 8354 0/0,
= 72° 20' 30", e=0.
The net efficiency, e, is therefore slightly less than the
gross efficiency, rj, for values of cf> nearly 90° ; but, as the
angle <\) diminishes, e falls rapidly below rj, and for
= 72° 20' 30" the energy dissipated is equal to the energy
stored,' and the net efficiency is therefore zero, while the
gross efficiency is about 38 per cent. For greater angles of
lag the loss is greater than the maximum energy stored, and
e becomes negative (see fig. 9).
For ordinary cases the angle <j) is greater than 88°, and e
and 7/ are nearly equal. Since the wattmeter method gives
directly the value of (f> it is much easier to express the value
of the net efficiency e (namely, 1 — 7r cot <£) than the value
of 7]. For small values of cot <£ this is sufficiently exact to
write it e = 1 — it cos (j>.
Suppose that instead of assuming the effective resistance r s
of the condenser to be in series with the condenser, as we
have done in figs. 2, 3, 7, and 10, we consider that it is in
Phil Mag. S. 5. Vol. 47. No. 284. Jan. 1899. D
34 Messrs. E. B. Eosa and A. W. Smith on a Resonance
parallel, as in fig, 11. Of course there is a slight leakage
current in every case, if the resistance of the dielectric is not
infinite. Boucherot* says of his paraffinpaper condensers
Fig. 11.
WWWWV 1
that the " heating is chiefly due to the Joule
effect, that is, to leakage current; the action
of dielectric hysteresis, if it exists at all,
being very slight." We shall give reasons in
a subsequent paper for believiug that this is
seldom, if ever, true of good condensers, but
at present let us assume it to be true. Then
the condenser current is 90° ahead of, and the
leakage current in phase with, the impressed
electromotive force. I being the total cur
rent, the condenser current is I sin <f> and the
leakage current is I cos <£. The energy
stored is
W = iCE 1 2 ,
and the energy dissipated per second is
JExIj cos <£,
or per halfperiod
™ = 4n ElIlC0S( ^
The maximum condenser current, I x sin <£, =pCE 1 .
w
_ E 1 2 .joCcos<j> _ 1
4n . sin (f> ~ 2
CExV cot (f>.
w
as before.
V 6=1 — ™.= 1 — 7T COt (py
Referring to fig. 13, we can derive anew the value of the
net efficiency. Curve 1 is the electromotive force, curve 2
is the current, in advance in phase by the angle (/>, nearly
90° ; curve 3 is the power, the positive and larger loop being
the work done on the condenser, and the negative and smaller
* VEclairage Electrique, Feb. 12, 1898.
Method of Measuring Energy dissipated in Condensers. 35
Fig. 13.
v:
36 Messrs. E. B. Rosa and A. W. Smith on a Resonance
loop being the work done by the condenser upon its discharge.
Equation°(l), p. 31, shows" the area of this powercurve to
consist of two terms, the coefficient of one containing I cos
and of the other I sin <£>. JsTow Icos<£ is the component of
the total current which is in phase with the E.M.F., and is
represented by curve 5. lsin<£ is the condenser current.
90° ahead of 'the E.M.F., and is represented by curve 4.
The powercurve for 4 is 6 ; its positive and negative loops
are equal, and it is the powercurve for a perfect condenser.
The powercurve for 5 is 7, and is the total work done, or
the energy dissipated. One loop of 6 is energy stored, W, one
w
loop of 7 is energy dissipated, w, and the ratio ^ is the
relative loss, or 1 — v^ is tne net efficiency.
The area of one loop of 7 is
= EJ^ r _ ^ipp j  = gj, cos f* =Energy pitted.
The area of one loop of 6 is
W= EA sin <f>  cos jptl^t EJl * in * . 2=Energy Stored.
.'. 6=1— ^ =1 — 7rcot 0, as before.
If the equivalent resistance of the condenser is taken to be
a series resistance then we have E cos <j) for the active E.M.F.,
E sin cj> for the condenser E.M.F., and the same result follows.
Example of the Resonance Method.
The condenser used was one which we had made ourselves,
and consisted of paraffined paper and tinfoil. The paper was
12x17 centim. and '0038 centim. thick; the tinfoil was
•0025 centim. thick, and its effective area was 10 x 15 centim.
approximately. The paper and tinfoil were piled up dry and
clamped between brass plates. It was then placed in melted
paraffin and maintained for some hours at 100° to 150° C.
This condenser then had a volume of about 300 cub. centim.
and a capacity of about *8 microfarad.
The resonancecoil consisted of 3000 metres of No. 10 wire
(B and S gauge, '259 centim. diameter) wound into a coil of
40 centim. internal diameter, 56 centim. external diameter,
and 17 centim. axial length. Its resistance was about 10 ohms
and its inductance 1*60 henrys. This coil was wound in ten
Method of Measuring Energy dissipated in Condensers. 37
sections, so that by choosing different sections or combina
tions of sections, a wide range of inductance could be secured.
In this particular case the entire coil was used. The fre
quency of the alternating electromotive force was varied by
varying the speed of the generator, complete resonance being
attained at a speed of 2175, for which the frequency is 145.
The current was 1*20 amperes, the resistance r c + r w was
982 ohms, I*{r a + rJ = 14'15 watts. The wattmeter gave a
deflexion of 188, corresponding to 37*6 watts. This leaves
23'45 watts for condenser loss, or EI cos <£.
E being 1808, 1 = 120, EI = 2169, and cos 0= ^ ='0108;
7r cot <j> = 3*39 per cent., e= 96'61 per cent.
The quantity of the eddycurrent loss in the coil does not
of course appear. From subsequent experiments we became
satisfied that it w r as large enough to cause a serious error in
the above value of the condenser loss. Hence we shall not
give any of the other values found using this coil. The
results obtained over a range of from 400 to 2250 volts
showed that the loss is sensibly proportional to the square of
the electromotive force. This conclusion is not seriously
affected by the presence of eddycurrents, since the latter are
themselves proportional to the square of the E.M.F. and yet
are not large enough to swamp the condenser loss.
We therefore wound up a coil of nearly 2000 metres No. 14
wire (B and S gauge, diarn. '160 centim.), in 41 layers of
45 turns each, external diameter of the coil being 37 centim.
The eddycurrent loss in this coil is less, owing to the smaller
diameter of the wire and the smaller quantity; a subsequent
measurement by an independent method gave 3*2 percent, as
the increase of the effective resistance by the eddycurrents
at a frequency of 120. At a lower frequency it would, of
course, be less. Its use will therefore illustrate the method
and give a fairly accurate value of the condenser efficiency.
Measurements on Beeswax and Rosin Condensers.
We give below a series of measurements on the efficiency
of a set of commercial condensers made of tinfoil and paper,
the latter being saturated with melted beeswax and rosin.
We understand that they are piled up dry, as we have done
with condensers made in our laboratory, and while immersed
in the melted beeswax and rosin placed in a receiver from
which the air is exhausted, to free them from air and moisture.
With the details of the process we are not, however,
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A Method of Measuring Energy dissipated in Condensers, 39
acquainted, and cannot say whether it is something in the
method of manufacture or the nature of the dielectric which
makes the dissipation of energy so large ; we presume, how
ever, that it is the latter. There were six condensers joined
together, each being a solid slab of about 11 centim. x 15 cen
tim. x 1*5 centim., thus having a volume of about 250 cub.
centim., and a capacity of one third of a microfarad.
The six slabs were placed on a table, joined together in
parallel, and in series with the resonancecoil (which was at a
distance from them and from the measuring instruments),
loosely covered with a woollen cloth, and coil and condenser
subjected to an alternating electromotive force of about
50 volts, and a frequency of 120. No effort was made to
secure the maximum degree of resonance, and the voltage on
the condensers was found to be about 900. In a short time
the temperature of the condensers had risen to 30°, as indi
cated by a thermometer inserted between two of them, and
the first set of readings was taken. The loss of energy in
the condensers was greater than it had been at lower tem
peratures, and continued to increase as the temperature rose.
At the same time, owing to this increase in the equivalent
resistance of the condenser, the resonance ratio decreased and
the current and voltage on the condenser decreased. The
loss at 36° C. is 50 per cent, greater than at 30° C, and is
approaching a maximum. At 39° it is 9*5 per cent., and the
fourth reading at sensibly the same temperature (but which
doubtless was a little higher, at least in some of the conden
sers) showed a slightly less loss. At 47° C. the loss had
decreased to 8*0 per cent., and at 49°'5 C. to 6*5 per cent.,
only twothirds its maximum value. The condensers were not
all at the same temperature, and the indicated temperatures are
therefore not exact. But they show unmistakably a maximum
value of the condenser loss, or energy converted into heat, at
about 39° C, and beyond that a very considerable diminution.
No further readings were taken until the condensers had
risen several degrees, when it was suddenly noticed that one
pair was hotter than the others and getting soft. The ther
mometer in a cooler pair registered 59°, but the warmest pair
was considerably higher. The loss was astonishingly large,
but the condenser had not broken down. Moreover, the
" leakage current " had not greatly increased, for while 839
volts gave 1'50 amperes at 49°' 5, 433 volts gave '80 ampere
at 59° C. To be in exact proportion to the voltage the current
should have been *774 instead of *80 at the higher tempera
ture, a comparatively small difference.
To find so large a loss in commercial condensers of good
40 X)r. E. H. Cook on Experiments
repute was a surprise to us. To find a well marked maximum
as the temperature rose, beyond which the loss decreased as
the beeswax and rosin composition softened was a second
surprise. To find so large a loss as the last observation
shows without the condensers giving way, and without any
very large leakage current, was a third surprise.
In order therefore to verify these results by a totally
different method, and to determine as accurately as possible
the losses in some paraffinpaper condensers which we possessed
which showed relatively very small heating effects, we built
a special form of calorimeter, into which the condensers could
be placed and the heat directly measured. The calorimeter
was copied after the large respiration calorimeter which one
of us designed for experiments under the patronage of the
U. S. Government, and which is located at Wesleyan
University. The description of the calorimeter and the
results obtained with it are reserved for a subsequent com
munication. We will only add that they fully confirm the
unexpected results obtained by the resonance method given
above concerning the dissipation of energy in beeswax and
rosin condensers.
Wesleyan University,
Middletown, Conn., July 1, 1898.
III. Experiments with the Brush Discharge.
By E. H. Cook, D.Sc. (Bond.), Clifton Laboratory, Bristol* .
[Plate I.]
rt^HE ordinary phenomena which accompany the brush
JL discharge are wellknown, but in view of the recent
extension of our knowledge of electric discharges in high
vacua, it seemed desirable to study the subject a little more
closely. The following experiments have been made with
this object.
Most of the results have been obtained with an ordinary
Wimshurst machine with 15inch plates, but they have also
been produced with the discharge from an inductioncoil, as
well as, though less readily, with a plate frictional machine.
In experiments requiring the production of the brush for
a short period the machine was turned by hand, but where a
longcontinued effect was desired, motion was obtained by
the use of one of Henrici's hotair motors. The number of
revolutions of the plates was counted by means of a tacho
meter, and the number of volts was taken as being about equal
* Communicated by the Author, having been read before the British
Association at Bristol, 1898.
with the Bru*h Discharge. 41
to what would be produced by the same speed of rotation
between knobs of one centimetre diameter. (See Joubert,
Foster and Atkinson, ' Electricity and Magnetism/ p. 103.)
In all cases the results have been produced at ordinary
atmospheric temperatures and pressures, but, of course, the
brilliancy of the effects varies with the climatic conditions.
For this reason no attempts have been made to measure the
size of the brush, because it differs so much from day to day.
The experiments described can be reproduced under varying
conditions, and the effects may therefore be regarded as normal
accompaniments of this kind of discharge.
1. Shape of the BrushDischarge.
As is well known, if a discharge of negative electricity takes
place from a pointed conductor, and if the point be examined
in a darkened room, it will be seen that it is surrounded by a
faint spot of light of a violet, or violetblue colour. If, on the
contrary, a similar experiment be made with positive electricity,
the point will be seen to be surrounded by an innumerable
number of lines of light of a similar violetblue colour, forming
what is called the brush. It is stated, on the authority of
Faraday, that the glow which surrounds the negative point is
separated from it by a dark space. Undoubtedly this is the
case when the discharge is taken in rarefied air, but at the
ordinary pressures I have been unable to detect it, although
the brush has been examined under the microscope. The
glow seems to be in contact with the point. The positive dis
charge, however, behaves differently. When carefully observed
it is seen that the lines do not start from the exact end but at
some slight distance away (2 or 3 millims.). They appear to
keep in a bunch for a little distance and then to diverge. The
size of the positive brush is much increased by the proximity
of an earthconnected plate or sphere, and the outline of the
luminous portion is altered by the shape and nearness of this
" earth." Thus, when an u earth " is some distance away, the
emanation from a point may take the shape of a fan with the
side lines at right angles to the point as in the figure. If to
such a brush an " earth " be brought to within a few centi
metres, the lines will curve themselves round and the angle
of the fan instead of being 180 degrees will become much
less. The glow at a point giving negative electricity becomes
brighter if an earthconnected body be brought near, but it
does not alter in size until the body is very close (less than a
centimetre), when small sparks pass between the point and
the body.
42
Dr. E. H. Cook on Experiments
The angle of the point makes a considerable difference to
the shape of the positive fan. If the end consists of a small
angle, for example a needle, the bounding lines of the fan
enclose a small angle, and the whole of the luminous portion
45 000
VOLTS
86.000
VOLTS
26 000
VOLTS
45/000
VOLTS
50.000
VOLTS
is very small. As the angle increases so does also the angle
of the fan. The figures show the kinds of discharge obtained
from brass points of varying angles. They were obtained
from the positive side of a machine, and under similar con
ditions of proximity to earthconnected plates. The dis
charges are drawn of about actual size, but the wires and
points are drawn larger than they were for the sake of
clearness. The approximate differences of potential between
the knobs of the machine when giving these discharges are
ivith the Brush Discharge. 43
stated. The appearance of the discharge obtained from
concave ends is also shown. Concave terminals behave like
angular ones.
2. Force of Wind from Points.
The mechanical force exerted by the strongly electrified air
particles which are repelled from the points is well known,
and the experiment of blowing out a candle is one of the
commonest shown as illustrating the action of points. The
magnitude of the force was roughly measured by causing
the discharge to play upon one pan of a delicate Robervahl
balance and measuring the weight necessary to restore the
equilibrium. Care was taken to use the same point and to
change the polarity by reversing the machine. This was the
only way in which comparative experiments were possible.
If the attempt be made to measure from two different points
it will be found that the minute differences in the points,
notwithstanding every care to make them precisely similar,
will show themselves by entirely altering the appearance of
the brush, and great differences in magnitude will be observed,
even with the same kind of electricity, from the different
brushes.
In order that vibration might be avoided as far as possible,
the measurements were made on a stone slab, built up from
the foundations of the laboratory. The following results
were obtained : —
Wimshurst machine 15 in. plate.
No. of sectors on each plate . . 16.
Size of sectors 8 sq. cm.
(1) Speed of revolution 450 per minute.
Potentialdifference corresponding to
this speed, about 43,000 volts.
Force from positive brush equal to
a weight of 0*29 gramme.
Force from negative brush equal to
a weight of 0*24 gramme.
(2) Speed of revolution 182 per minute.
Potentialdifference 35,000 volts.
Force from positive brush equal to a
weight of 008 gramme.
Force from negative brush equal to
a weight of 0*066 gramme.
The best distance of brush from pan
of balance 0*04 metre.
44 Dr. E. H. Cook on Experiments
Induction Coil : —
Length of primary wire . 200 feet (500 turns) .
Length of secondary wire . 60,000 feet (490,000 turns).
Capacity of condenser . . 2 microfarads.
Length of spark in air . . 0*07 metre.
Corresponding to
* Potentialdifference of about . 63,000 volts.
Force from positive brush equal to a weight of
0*01 gramme.
Force from negative brush equal to a weight
of 0'01 gramme.
It will thus be seen that the magnitude of the force is
greater from the positive side of the machine than from the
negative.
An attempt was now made to find the maximum distance at
which this mechanical disturbance could make itself felt.
For this purpose the following experiment was arranged. A
single fibre of unspun silk was stretched across the field of
view of the microscope, and on this was hung a little paper
index which was blown aside by the wind from the point.
The maximum distance at which any deflexion could be
observed was then noted, and in this way a comparison was
instituted. Care was again taken not to use different points,
but to alter the polarity by reversing the machine. Also,
during the experiment, the whole apparatus was carefully
protected from extraneous currents of air. It was found that
with the machine with a potentialdifference of 33,000 volts
the positive brush produced an effect at a distance of 0'6 metre,
the negative at a distance of 0*48 metre. When the potential
difference had fallen to about 25,000 volts the distances
observed were for the positive 0*32 metre, and for the negative
028 metre. With the coil giving a spark of 4 centimetres
(41,500 volts) the positive brush affected the thread at a
distance of 32 metre, and the negative moved it at the
same distance.
These results therefore confirm the former ; for the positive
brush affected the silk at the greater distance, and also pro
duced the greater pressure upon the pan of the balance. But
they show how very quickly a moving electrified particle of
air is brought to rest by surrounding air.
* These details are given in order that an idea may be formed of the
kind of apparatus worked with, and that if any one should desire to
repeat the experiments he may know what results to expect. The
actual numbers will vary with the apparatus.
with the Brush Discharge. 45
3. Electrical Action at a Distance.
If an electroscope or a leydenjar be placed at some distance
from a point from which a brushdischarge is taking place,
it will become charged. If the brush be a positive one the
electroscope or jar will be charged positively ; if the brush be
a negative one it will be charged negatively.
The distances at which the effects make themselves evideut
vary with the potential and the atmospheric conditions.
Statements of lengths therefore are only valuable as allowing
of comparison being made between experiments which are
performed at the same time and in the game laboratorj^. The
same relation will, of course, hold, but the actual measure
ments will be different. With these reservations the following
are given : —
With the plates of the machine revolving at the rate of
450 revolutions, and giving a difference of potential of about
43,000 volts, an electroscope was affected at a distance of
1'8 metres from the point. No difference was made by altering
the polarity.
When the plates were revolving at the rate of 105 revolu
tions, and the potentialdifference was 25,000 volts, the
same electroscope was affected at a distance of 1/0 metre.
The size of the collectingplate of the electroscope makes a
considerable difference in the ability of the instrument to
become charged. The larger this plate the greater the
distance at which it can be charged. A point on the plate
reduces the distance somewhat, probably because it allows
the electricity to escape as fast as it can be collected.
No increased effect could be obtained by increasing the
condensation.
The shape of the point which gives the best results is
acute. This is different from the effect in producing lumi
nosity. In that case it will be seen that a larger luminous
brush is obtained with a greater angle up to 90°.
Notwithstanding the enormously greater difference of
potential produced, the brushes from the coil used were quite
unable to influence the electroscope at distances equal to those
obtained from the machine. Thus, when the coil was giving
sparks of 0*07 metre in length and thus causing a potential
difference of about 63,000 volts, the positive brush was only
capable of charging the electroscope at a distance of 0*62 metre
and the negative at the same distance.
The same relative results were obtained when the brushes
were used for charging a leydenjar. Thus, with a difference
of potential of about 40,000 volts, a jar was charged by the
46 Dr. E. H. Cook on Experiments
positive brush from the machine at a distance of 0*55 metre.
The negative brush charged it negatively at the same distance.
With a concave terminal this distance was reduced to
0*50 metre.
With the coil giving a spark of about 0*7 metre (63,000
volts) the jar was charged at a distance of only 0*14 metre.
But, as has been already stated, these distances vary greatly.
With favourable atmospheric conditions, and a plate on the
electroscope 16 by 23 centims., it was found that the electro
scope could be readily affected at a distance of 4 metres from
the point attached to the 15inch Wimshurst machine. On
comparing these distances with those obtained when measuring
the mechanical force of the wind from the point, it is seen
that the electrical effects are felt at a much greater distance,
amounting, if we take the maximum distances, to more than
six times.
It is not at all necessary that the conductor ending in the
point should be directed towards the electroscope. The
instrument will be found to be influenced in almost any
position ; in fact, when experimenting in frosty weather, I
have obtained results at equal distances all round the point,
even in the exactly opposite direction. It is therefore clear
that the point is the centre of a disturbance which radiates
from it in all directions.
The interposition of objects between the point and the
electroscope gives interesting results. A wire cage, if com
pletely covering the instrument and earthconnected, will
prevent it from being charged, but if it is not completely
covered, even if only one side of the base is tilted up, the
instrument will become charged. The same wirecage held
between the point and electroscope has no effect. Plates of
metal, wood, or other material placed between, do not alter
the effect unless they are either close up to the point or to the
electroscope. In both cases a diminished effect is observed.
Experiments were tried with the view of discovering if the
effects could be produced through substances ; i. e. if like the
Rontgen rays the brushdischarges possessed any penetrative
power. Although by this mode of experimenting definite
proof could not be obtained, several of the results are instruc
tive. Thus, if the electroscope be placed at from 2 to 3
metres away from thepoint, and a board 56 by 78 by 15 centims.
be interposed midway in the path of the discharge, the leaves
are easily affected, almost as easily as if the board were absent.
The same thing happens if sheets of metals are interposed.
The direction of the point does not affect the result. If the
sheet be placed within 30 centims. of the point, the effect on
ivith the Brush Discharge. 47
the leaves is not produced. If the sheet be placed near to
the electroscope {within 10 centims.) no effect is produced on the
leaves, but as the distance is increased the leaves diverge more
and more ; thus at 50 centims. the effect is nearly as much as
when the board is midway. It is impossible while witnessing
these experiments to avoid calling to mind the similarity of
the effects produced to those which one sees on the coast when
a long billow rushing onwards towards the shore meets with
a solitary rock in its path. The rock is grasped on all sides,
but while immediately behind it the water is comparatively
still, the waves soon curl as it were round the sides and meet
each other at a short distance behind. The result being that
the effect of the rock is to interfere with the wave for only a
short distance immediately behind it. Beyond that there is
as much commotion in the line from the middle of the rock to
the shore as at the edges.
If while the electroscope leaves are diverged and the
machine working, a person walks between the point and the
instrument, the leaves will sway with his movements, but will
not fall together.
The production of the effect all round the point extends to
positions under the machine, for I have repeatedly obtained
the divergence of the leaves when the electroscope was placed
directly under the machine, which stood on an inch board on
the bench, which had a twoinch top.
4. Chemical Action produced by the BrushDischarge.
The formation of ozone by the working of an ordinary
electrical machine is well known, but whether positive or
negative is more active in its production has not been
investigated. It was therefore determined to examine the
action of the brushdischarge in producing this and other
chemical changes. The whole of the work contemplated has
not been completed, but some results of interest may be
mentioned. Considerable difficulty was experienced, in con
sequence of the high potentials used, in preventing leakage.
In some cases leakage took the form of brushes at other
places than the points required. In such cases the results
were useless. In consequence of this, whenever it was
necessary to lead the brush away from the point only the
most strongly insulated wire was used, and this was first
specially examined in order to see if any cracks or thin
places were in it. Whenever the wire had to be led into
flasks or bottles it was found that the best material for bungs
was solid paraffin. But it was found very difficult to produce
good brushes inside glass vessels, because the interior surface
48 Dr. E. H. Cook on Experiments
very soon became rapidly charged, and this seemed to pre
vent the production of the brush.
Experiments with Potassium Iodide.
The chemical action produced by the brushes in air may be
the formation of ozone, the production of oxides of nitrogen,
and perhaps other less known combinations. Potassic iodide
would be decomposed by ozone and the oxides of nitrogen,
and it is therefore a suitable substance to experiment with.
After many trials the simplest apparatus was found to be the
most useful. The points attached to the positive and negative
sides of the machine were placed at such a distance above the
surface of some standard potassic iodide solution that no
actual spark from the point could pass to the solution (with
the machine and potential used, this was anything beyond
2 centim.). This precaution was taken in order that no breaking
up of the iodide should take place in consequence of the
spark passing through it. The solution was contained in glass
or porcelain dishes, and as solution of potassic iodide (unless
perfectly pure) becomes slightly coloured when exposed for
some time to air and light, a similar quantity of the standard
solution w 7 as placed in an extra dish of equal size at the same
time. At the conclusion of the experiment the amount of
iodine set free from this blank was estimated and subtracted
from that produced under the brushes.
The results obtained have differed among themselves as far
as actual amounts of iodine set free at one brush compared
to the amount set free at the other, but they have all agreed
in this important particular, that the amount of iodine produced
by the negative brush from the machine is always very much
greater than the amount produced by the positive in the
same time.
With points prepared as similarly as possible, and every
precaution taken to avoid leakage, it was found that from, five
to eight times as much iodine was set free by the negative
brush as by the positive. The following are details of a
typical experiment : —
Speed of rotation of plates of machine, 300 per minute.
Potentialdifference about 40,000 volts.
Distance of points from surface of solution, 4 centims.
Amount of iodine liberated by the negative brush over that
in blank in halfhour, 0*000762 gramme.
Amount of iodine liberated by the positive brush over that
in blank in halfhour, 0*000 127 gramme.
The greatest amount of iodine liberated in one hour in this
way amounted to 0'001778 gramme ; assuming this to be
with the Brush Discharge. 49
produced by the formation of ozone, which then decomposes
the iodide, it would correspond to the formation of
0'000112 gramme of ozone.
Condensing the electricity produces very little effect. For
with the jars on, similar results have been obtained to those
without the jars. Thus with points 4 centims. away and a
potentialdifference of about 40,000 volts, 0*001778 gramme
of iodine was produced at the negative brush, and 0*000190
gramme at the positive in one hour.
A diminution of potentialdifference reduces the amount of
chemical action ; but the substance of which the point is
composed is immaterial. Thus, with 25,000 volts difference,
0*000698 gramme of iodine was set free by the negative in
half an hour, as against 0*000762 gramme with 40,000 in the
same time.
The distance of the point from the surface of the liquid
makes an important difference. It has already been stated
that all the experiments were made when the points were
farther away than that at which a spark could pass. The best
effects are caused when the points are just farther away than
this. Thus when the points were 12 centims. away 0*000021
gramme was liberated by the negative, as against 0*000063
gramme when 8 centims. away, and 0*000508 gramme when
4 centims. away.
Similar experiments to the above made with the brushes
obtained with the coil have given different results. In this
case it was found that invariably the brush from the positive
terminal liberates the greater amount of iodine. The difference
between the amounts produced by the positive and negative
brushes is not, however, so great as that by the machine. In
this case also we find differences between individual experi
ments, but as before always an excess by the same pole.
The average of many experiments gave from three to five
times as much iodine set free under the positive brush to
that set free under the negative. These numerical results
apply to the coil whose dimensions are given above.
It was found that with the machine and coil as described,
a much greater quantity of decomposition was caused by
brushes from the machine than from the coil. This corre
sponds to the amount of visible brush produced.
Actum on other Substances.
The power of the brushes to produce other chemical
actions has been partially investigated. So far no reducing
actions have been observed, but oxidizing ones are always
present. In these cases with the machine brushes more action
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. E
50 Dr. E. H. Cook on Experiments
takes place at the negative than at the positive, and generally
the same conditions apply as in the case of potassic iodide.
Thus about four times as much iron is oxidized from the
ferrous to the ferric condition by the negative as by the
positive brush.
5. Effect on the Electrodes.
It is stated* on the authority of Wheatstone that : —
" Metallic dust is in every case torn away from the electrode
by the brush discharge/' This statement is one which would
be supposed to be true when it is considered that such is
known to be the case with the sparkdischarge. But it is
contradicted by spectroscopic evidence, for the spectrum of
the glowdischarge shows no trace of metallic lines. It is the
same u whatever the nature of the metal, and is due solely to
the incandescent gas/'f In order to test the statement
several experiments were made. The brushes were obtained
from copper points, and made to play upon the surface of
some dilute nitric acid placed in dishes under them. The
points were brought as near as possible to the surface, i. e.
as near as possible without producing a sparkdischarge.
They were so near that minute waves were formed during
the whole time of the experiment by the wind from the points.
No possible loss of copper could therefore occur if any were
torn off from the point. The machine was run at its highest
speed, and thus the greatest difference of potential available
(from 40,000 to 50,000 volts) was obtained. The experiment
was continued for two hours, during the whole of which time
very fine luminous brushes were being produced at each point.
At the conclusion of the experiment the acid was carefully
concentrated by evaporation and tested for copper. Not a
trace could be found in either. Thus showing that no metallic
particles icere torn off the points. Moreover, I have examined
microscopically a pair of points which I had cut and prepared
from a piece of guttapercbacovered wire. These points have
been continuously in use for some months, and it is certain
that at a low computation brushes must have been drawn
from them for at least 150 hours. The edges seem as sharp
as when cut, and sensible alteration has not taken place in
them.
Of course these results apply to the potential and quantity
worked with. Higher potential and increased quantity may
give quite different results.
* Silvaims Thompson's l Elementary Lessons/ p. 305.
t .Toubert, Foster and Atkinson, ! Electricity/ p. 103.
with the Brush Discharge . 51
6. Action of the BrushDischarge upon Photographic Plates,
The productions of chemical actions by the brushdis
charges immediately gave one the desire to try the action
upon photographic plates. The first experiments of this kind
were made shortly after the discovery by Rontgen ; but as
they did not lead to any results which were of such general
interest as the latter, the study was discontinued for a time
but taken up again recently.
In these experiments, as in the former, the brush from the
machine gives better results than that from the coil, but the
positive seems to be more effective than the negative. In
the latter case a longer exposure is necessary, and sometimes
less definite results are obtained. Films of different kinds
were used, but because of the greater ease in development,
ordinary " slow " plates were most frequently employed
("Ilford Ordinary").
Action of BrushDischarge upon a Sensitive Film.
If a photographic plate be placed on the table in a dark
room with the uncovered film upwards, while the positive or
negative brush from a machine or coil be arranged at some
distance (say 4 feet) above it and the point turned towards it ;
the plate after development will be found to be " fogged,"
showing that a decomposition of the silver salts has been
brought about, similar to that produced by exposure to light.
With the brush from the positive terminal the reduction of the
silver compound is fairly uniform all over the plate, but when
the negative is used there are " blotches " in several places,
showing that the reducing action has taken place in some
spots more than in others. There is therefore produced at
the point an emanation of some kind, whether it be an undu
latory movement or a stream of particles, which possesses the
power of reducing silver salts.
If between the point and the sensitive film a solid object
be placed, a shadow of the object will be thrown upon the
plate. This shadow is sharp if the object is close to the
plate, but its edges are illdefined if it be at some distance
from it, in this particular exactly resembling light. PI. I.
(figs. 1 and 2) shows this, the object being a piece of sheet
zinc cut into the shape of the letter H and placed firstly at
1*5 centim. away from the plate, and secondly at 10 centims.^the
point being about 30 centims. away and the potential about
35,000 volts. By measuring the size of the object and images,
and the distance between them, it is easy to find the position
of the source from which the rays emanate. When this is
done it is found that the rays proceed from the extreme end
E 2
52 Dr. E. H. Cook on Experiments
of the point and not at some distance from it. The visible
divergence of the brush does not start exactly from the point
but at some little distance in front of it. It would seem
therefore that the point of the emanation of these rays does
not coincide with that of the visible part of the brushdis
charge.
The next step was to find if the effect could be produced
after reflexion. This was shown with the following apparatus.
A cardboard box was made of the shape shown in sketch.
i§
A B is a mirror, the positive terminal of the machine, D an
object, and E the photographic plate. Internal reflexion was
carefully prevented and the experiment made in the dark
room. The point was about 20 centims. from the mirror and the
mirror about the same distance from the plate. The object,
which was a piece of sheetzinc, was from 1 to 5 centims. from
the plate. When the machine was worked, and no mirror in
position, it was found that the exposed plate was unaffected,
the rays we may imagine being absorbed by the sides of the
box. This experiment also shows that the fogging of the
plate is not caused by any ozone which may accompany the
discharge. When, however, the mirror is in position a dis
tinct image is formed upon the plate after a slight exposure
(5 minutes was usually given). Supposing the effects to be
caused by the emission of electrified particles from the terminal
it might be imagined that these would be reflected from the
mirror and so turned from their course as to impinge upon
the plate. In order to discover if this explanation be correct,
the mirror A B (which was of glass in the first experiment)
was replaced by a sheet of metal. Xow, if electrified particles
fall upon this and it be earthconnected, their electricity will
with the Brush Discharge, 53
be immediately discharged, and therefore when they subse
quently impinged upon the plate no effect would be produced.
This was found not lo be the case, as is shown by PI. I. (fig. 3),
which was produced by employing a piece of tin for the
reflecting surface instead of glass.
That the metal plate did receive the impact of electrified
particles was shown by the fact that sparks could be drawn
from it from time to time.
In view of the statement of Tesla, that the efficiency of
the metals as regards their reflecting powers for #rays
follows their order in the voltaic series, an attempt was made
to test if any variation of reflecting power could be detected
in the present case. For this purpose the metal reflecting
plate was varied, but the experimental results show that no
conclusions can be drawn indicating any connexion between
the reflectingpower and position in the electrochemical series.
In fact the reflexion seems to depend upon the brightness of
the surface, and therefore agrees exactly with the reflexion
of light.
7. Penetrative Effects produced by the BrushDischarge.
The remarkable results obtained by Hontgen and others
induced an attempt to imitate the effects by the brushdischarge.
For this purpose a sensitive plate was wrapped in brown
paper (two folds), and on the paper were placed sundry small
articles, such as coins, keys, &c, and the whole exposed to the
brushdischarge. The experiment was made in a darkened
room, and the point placed at about 5 to 6 centims. above the
coins, the plate lying on the table. An exposure of 30 minutes
was given, and on developing the plate the outline of the
articles was distinctly shown. This experiment was repeated
with brushes of different polarity and source {i. e. coil and
machine), and in every case the same results were obtained.
Probably the cause of this action is that the substances
become charged and act inductively upon the silver salt in the
sensitive film, causing a partial decomposition or production
of a " latent image/' which decomposition is carried still
further in the process of development of the plate. The next
step was to see if the outline of the bones could be produced
without the flesh. Numerous experiments were made, but
although in all cases the outline of the hand could be repro
duced in no case did the bony skeleton show itself.
None of these effects could be obtained when a piece of
vulcanized fibre was included in the wrapping. This substance
has been shown by Giffard and others to be impervious to
#rays, and, of course, it is equally impervious to light.
54 Dr.E.H.Cook on Experiments
As the thickness of the wrapping increased, definition on
the developed plate became less and less. Thinking the effect
might be due to the fact that the brown paper wrapping was
not lighttight experiments were made to test this idea, and
it should here be mentioned that the brown paper referred to
is that which is used, by photographicplate makers to wrap
sensitive plates in. Firstly, a plate wrapped in one fold was
exposed for one hour to the light of an ordinary 8 c. p. glow
lamp illuminated by a 105 volt alternating current. A negative
was obtained clearly showing images of the objects placed on
the brown paper. This seemed to support the idea. Then
a wrappedup plate was simply exposed to diffused daylight
for two hours. A very faint and blurred image of the coins
&c. was obtained. Thus showing that the paper was not
absolutely lighttight. Moreover, by exposing to very power
ful light, such as that from burning magnesium and the
limelight, clearer effects were obtained, showing that the
more powerful light is capable of getting through the paper
better.
8. Reproduction of Prints fyc. by BrushDischarge.
Whilst engaged in obtaining a perfectly lighttight
wrapping for the plates it happened that a piece of ordinary
notepaper was used and the whole exposed to the action of
the brush. On developing the plate a clearlydefined image of
the watermark of the paper was produced (fig. 4) . This induced
trying to copy in a similar way printing, writing, pictures,
&c. In every case this has been done with complete success.
If a photograph, or a drawing, or printing, or writing be placed
in contact with the sensitive film and exposed to the brush
discharge, a clearlydefined and very sharp reproduction is
obtained. This seems to be more readily produced by the
positive brush than by the negative. The effect on the plate
can also be produced if the drawing be not in contact with
the film but separated from it by one or two layers of paper
or cardboard. In the latter case, however, the definition is
not so good.
The effects produced when the drawings are in contact
cannot be produced simply by keeping the plate and print in
contact, at any rate for the same time as was used in my
experiments, but can be brought about by exposing the plate
to a powerful light, providing the wrappings are not too
many.
One of the results of these experiments, of a somewhat
startling character perhaps, is that the writing on a letter
inside an envelope can be reproduced. It may be some
loith the Brush Discharge. 55
comfort, however, to know that there is considerable difficulty
in recognizing the words owing to the folding of the paper,
and thus one word coming immediately on the top of another.
Fig. 5 shows the result obtained when a printed invitation
card enclosed in an envelope was exposed to the brush
discharge, the envelope being placed on the sensitive film.
Notice the texture of the paper, and the opacity caused by
the gum and double thickness of paper.
Light from the Discharge.
The great similarity between the effects recorded in the
foregoing pages and those produced by ordinary light led to
experiments being made to compare them,
The first point which suggested itself was to see whether the
interposition of a body which was transparent to ordinary
light, between the point and plate, made any difference in the
result. A plate of clear glass was used, and the shadow of
an object obtained. No difference was observed in the sharp
ness of the image. The glass was now blackened with lamp
black until it was so opaque to light that the flame of an
ordinary candle could not be seen through it when it was
held at a distance of 5 centims. from the flame. When this
was held between the point and the photographic plate no
effect whatever could be obtained.
Again, the law of inverse squares was proved in the follow
ing manner : — A small cross cut out of thin sheetzinc was
placed at a certain distance from the sensitive plate and its
shadow obtained. The plate was now moved so that the
shadow should fall upon a different part of the film, and the
object was placed at a different distance away. Another
shadow was now obtained. This was of a different size to
the former. The two figures were then measured, and the
sizes compared with the object and the distance of the brush.
These results therefore indicate that the effects are produced
by the light which the brush emits. Moreover, after many
trials, I have been enabled to reproduce all the effects with
artificial sources of light. Many of them can be produced by
employing daylight, and probably all could thus be formed,
but the length of exposure required has hitherto prevented
this from being done.
But, notwithstanding this apparently simple explanation, it
is quite possible that we have something else taking place at
the same time.
Suppose that the point is the centre of a disturbance from
which waves are emitted. These waves will be of various
lengths, some capable of affecting the optic nerve, and some
56 On Experiments with the Brush Discharge.
of shorter and some of longer wavelength than these. The
reduction of the silver salts in the sensitive film may be
caused by more than one kind of these waves. Before then
it is possible to say that the effects are caused by the actinic
power of the waves of short wavelength only, it is necessary
to separate these waves in some such manner as that employed
by Tyndall to separate heatwaves from those of light.
Various experiments have been made to do this, but hitherto
without success. The investigation is being pursued in this
direction.
The actual amount of light given out by the brush discharge
is not much, and in the reflexion experiments which have
been described it was found impossible for the most delicate
eye, even after being kept in absolute darkness for a con
siderable time, to distinguish the outline of the object; but the
shadow was nevertheless easily produced on the photographic
plate. The numerous experiments which have been made
show that, so far as one can judge, the effects produced by
the brush are far more definite than would be expected, when
the very small luminosity of the discharge is remembered.
Or, in other words, that the emanation from the point contains
a much larger proportion of rays capable of bringing about
chemical decomposition than would be supposed when we
remember only its luminosity.
In order that some approximation might be obtained between
the lightgiving power of the discharge and ordinary light, the
following experiment was made. A comparison was first ob
tained between the light from a standard candle and the smallest
burner procurable, the gas being burnt under the usual con
ditions for regulating the pressure. This burner was now
compared with the light from the brush. The ordinary Bunsen
photometer, as used in gastesting, was employed, and the
light from the machine carefully screened off. In order that
a good brush might be obtained an u earth " was placed near
the terminal. The actual figures obtained varied with the
climatic conditions, but not so much as would be expected,
and the following numbers may be taken as an approximation,
but, of course, only an approximation, to the relative lumi
nosity of a standard candle and the brush from the machine
used.
Distance of candle from photometer ... 60 inches.
Distance of light „ „ ... 3' 2 5 inches.
Therefore, Candle : Light : : 3600 : 10*56.
Distance of light from photometer . . .56 inches.
Distance of positive brush from photometer . 4 „
Therefore, Light from positive brush : Candle : : 1: 267,200.
Irregularities of Meteorological and Magnetic Phenomena. 57
The light from the negative point is less than that from the
positive. Brass points were used when making this com
parison, but no difference was observed with other points.
The potentialdifference was about 30,000 volts.
Comparison of the Actinic Poiver of the Brush with that
of Light.
In order to roughly test this an ordinary negative was
taken and fixed in a frame together with a piece of bromide
paper. The frame was then exposed in the dark room to the
light from a standard waxcandle. The same negative was
then treated in the same way and exposed to the brushdis
charge for a given time. The prints were then fixed and
compared in order to see if they were over or under exposed.
The experiments were then repeated until the effects were
equal. After many experiments the nearest comparison that
could be obtained was that the light from the candle at a
distance of 15 centim. and 20 seconds' exposure produced
the same effect as that from the positive brush at the same
distance and 15 minutes' exposure. From this the relative
powers would be as 1 to 45. The potentialdifference when
making these tests was about 30,000 volts.
On comparing this number with that obtained from the
photometric experiments it will be. seen how widely they differ,
and it may therefore be considered as certain that the emana
tion from the point consists very largely of those waves
which are capable of bringing about chemical changes.
IV. On the Analogy of some Irregularities in the Yearly
Range of Meteorological and Magnetic Phenomena. By
Dr. VAN RlJCKEVORSEL *.
[Plate II.]
AT the Toronto Meeting of the British Association f I
called the attention of Section A to the fact that if the
normal temperatures for every day of the year are plotted
down in a curve, such curves are strikingly similar for
stations spread over a very large area. An area which is
larger than our continent for some of the particulars shown
by these curves, while for others it extends over Western
Europe only, or over part of it.
I am now able, to a certain degree, to give an answer to
the query at the end of that paper : " Is it temperature alone
of which the irregularities are so extremely regular ? How
* Communicated by the Author.
t Paper published in the Phil. Mag. for May 1898.
58 Dr. van Rijckevorsel on Irregularities in the Yearly
does the barometer behave ? Do the winds, do the magnetic
elements, show something pointing to a common origin ? "
The answer is most certainly a positive one, as I hope to
show now.
On the diagram (Plate II.) the uppermost curve, marked T,
shows the temperature of every day of the year for the Helder
in the Netherlands. The next curve, marked H, shows the
horizontal, and the following one Z the vertical component of
the earth's magnetic force as shown by the registering instru
ments at Utrecht. The curve marked R the rainfall at the
same station. The curve marked P shows the barometric
pressure at Greenwich ; the last curve, D, the magnetic
declination at the same station.
T shows the mean of 50 years' observations, H of 33 years
between 1857 and 1896, Z of 30 years in the same period.
For the magnetic data it is not always possible to use all the
years for which values are available ; for if in any case a
serious break occurs in a series, which cannot be safely
bridged over by fictitious values, or if a discontinuity occurs
such as may be occasioned by a slight alteration in the instru
ment or in the adjustment of the scale, it is, as a rule, necessary
to reject that whole year — this for more than one reason, but
chiefly on account of the secular variation which is so irregular.
It is unnecessary, however, to explain this now at any length.
E, is the result of the 40 years 185695, P of the 18 years
discussed by Mr. Glaisher *, viz. 184158, and lastly "D of
the 30 years 186594, of which 1 owe the three years which
have not been published in the publications of the Observatory
to the courtesy of the AstronomerRoyal.
Of course not all these series are of the same quality; some
are decidedly not long enough ; but although a large amount
of material is still being computed, I am at present only
able to show T what there is in this diagram. The 19 years
for airpressure are decidedly not enough, and at the same
time all those who are interested in terrestrial magnetism can
form a judgment for themselves as to the intrinsic worth, of
the data for the vertical component. But for my present aim
the material is about sufficient.
All the curves have been smoothed down in the same
manner as explained in my first paper on this subject. For
the magnetic curves, however, a preliminary operation was
thought to be advisable, because the effect of the secular
variation throughout the year would otherwise tilt the curves
and render them less easily comparable with the others. This
very simple operation was to take the difference between the
* 9th and 10th Reports of the British Meteorological Society.
Range of Meteorological and Magnetic Phenomena. 59
mean for the ten first days of the year and that for the ten
last days, and to interpolate this difference over the days of
the year. The secular change has been taken into conside
ration in no other way; so that the figures from which these
curves are drawn are simply the mean of the scalereadings
for a certain number of years. Except for my purpose, they
have therefore no value whatever.
The scales on which these six curves are drawn are also
purely arbitrary. For H and Z even the scaledivisions
as published in the annals of the Koninhlijk Nederlandsch
meteorologisch Instituut at Utrecht have not been reduced to
any of the customary magnetic units. The factors by which
the original figures have been multiplied in order to form the
ordinates of these curves, ranging from 4 to J, were simply
chosen so that the consecutive maxima and minima of the
different curves should be, on an average, approximative^ of
the same importance. In other words, in the vertical dimen
sions the curves have been compressed or expanded so as to
render the phenomenon I wish to show as prominent as
possible.
And I think it will be admitted that it is prominent. Of a
few doubtful points I will speak later on. But if we except
these for the present, I think that it is perfectly apparent that,
however dissimilar the general behaviour of the individual
curves may be, every single maximum and every minimum in
one curve is, with an astonishing regularity, repeated in all
the others. Far from it that the curves should be parallel.
In many cases a maximum in one curve occurs earlier or later
than in most of the others. In still more cases, what is a
large bump or a deep valley in one curve may find its corre
spondent in a hardly perceptible movement in another curve.
A few of these differences in behaviour shall be, later on,
explained away in a satisfactory manner. Others may be due
to insufficience, as yet, of the data at command. But even
if a certain — decidedly small — number of instances should
remain where one of the curves really has a secondary maximum
and accompanying minimum not shown by the others, why
should it not ? It may be a local anomaly; it may be that
indeed for, say, the barometer an agent enters also upon the
scene which has less grip, or none at all, upon the thermometer
or the magnetometers. There are certainly not many such
exceptions to the general phenomenon shown in the curves
under discussion. Indeed, they are so few that I incline to
think that we may predict even now that as soon as we shall
be possessed of a sufficient number of sufficiently good obser
vations in every case it will indeed be found that to every
60 Dr. van Rijekevorse] on Irregularities in the Yearly
irregularity in one yearly curve, either for any magnetic or
any meteorologic phenomenon, a corresponding irregularity
will be apparent in all the other curves for the same region,
if not alwavs. as will be shown, for the same station. But
even if some doubtful point should remain. J. think proof is
so abundant even now that, even if I should be found ulti
mately to have overrated it in some instances, this can no
more invalidate the principle.
Another very valuable test for my opinion that the ana
logies pointed out here cannot be accidental, is furnished by
Ihis diagram for Paris *. The uppermost line shows the
temperature as resulting from 130 years' observations. The
lower one shows the magnetic declination, but this is the
result of only 12 years of observation between 1785 and 1796
bv Cassini. Xot verv good observations we should now call
them : there are several breaks and changes of continuity
which I had to bridge over as best 1 could. Moreover, the
readings were not taken rigorously at the same hour of the
day. 1 think this is more than sufficient, with such a short
series, to account for a certain number of doubts and queries
which I cannot explain away. But upon the whole 1 think
it will be admitted that the concordance between the two curves
is inst as satisfactory as in the large diagram. Therefore, at
least also in this respect, things a century ago were what they
are now.
One possibility I must point out. It may be that one or
more of these curves ought in reality to be inverted. This
seems preposterous. Still, with so many maxima and minima
following each other in rapid succession, it is not so. (I have
numbered 17 maxima; but I am perfectly sure that a larger
number than this are just as real as these, only they are so
small as not to be detected yet, as long as we are obliged to
smooth down the curves to such an extent.) Take a maximum
and the following minimum, which slightly precedes the cor
responding feature of the other curves, and invert the curve :
they will then simply be converted into a maximum and a
preceding minimum slightly lagging behind a similar feature
in the other curves. One of my magnetic curves is inverted
— needless to say which, because a scale may read from right
to left or from left to right. But also I have had serious
doubts as to the advisability of inverting one or two others.
You will see at once that in some cases my vertical lines f
* This diagram is not reproduced, as the text is sufficiently clear
without it.
f Continuous verticnl lines connect the maxima, lines of long dots the
minima belonging together.
Range of Meteorological and Magnetic Phenomena. 61
indicate a maximum where, in good faith, such a thing is not to
be discovered. This is quite true. I call attention to the
lines 6, 13, 14, 15. There is certainly no, or hardly an,
apparent maximum in the temperatureline there. But if we
look for a moment at this larger collection of temperature
curves*, of which it is not possible to give such a large diagram,
it will at once be seen that for the three last of these numbers
the curves for the whole of the United Kingdom show a small
but decided maximum in all these cases. For 13 this also
appears in some curves for southern stations, such as Mont
pellier. On the other hand, at 6 the maximum which the
curve for the Helder does not show is most distinctly shown
by the curves of the stations to the south and east of the
Netherlands.
There is, I think, a most valuable use of the method here
explained. For one phenomenon any of the maxima may be
less apparent than for another. It may even be not at all so,
(It would seem as if magnetism were a more sensitive organ
than meteorology.) But as soon as some of the curves show
a certain maximum, there is some presumption that the others
ought to do so, even if they do not. In other words, that
there is at that particular moment an influence at work which
would manifest itself by creating a maximum in those curves
also, if only our series of observations were long enough or the
methods of observation sensitive enough. If at some moment
of the year the vertical intensity or the rainfall show a
maximum or minimum, even an unimportant one, the tem
perature and declination must show it too — it may be smaller
or larger, or somewhat larger or later, but it must be there.
And should it decidedly not be there, depend upon it that here
is something worth investigating.
But again the point now under discussion gives us a
valuable clue to the direction in which we must look for the
origin of the anomalies of our curves ; and this may lead
perhaps in some cases to a guess as to their causes. From
what has just been said it is of course very probable that the
maxima in 14 and 15 are due to the influence of some cause
which has its seat to the north or northwest of the Netherlands,
the one in 13 to one in the southwest, but that in to an
influence coining from the southeast.
I must here refer again to a very remarkable minimum
occurring in some curves on or about the 1st of July, which
has been mentioned in the Toronto paper f. The mass of
material which has since come into my possession has shown
* These diagrams also are not reproduced,
t Phil. Mag\ May 1898, p. 405.
62 Dr. van Rijekevorsel on Irregularities in the Yearly
me that this is indeed more widely spread and more enigmatic
than I at first supposed. However, I think I am able now
to point out with a certain degree of probability where it
originates. This is the first case in which I really could do
more than show the direction in which this origin ought to be
found.
Arbroath
Valencia
Eothesay
Ihorshayn
Berufjord
Greenwich . .
Vlissingen ..
TheHelder ..
Christiansand
This second diagram shows for a certain number of stations
the daily mean temperature for the mouth of July and parts
of June and August. The order in which the curves are
arranged is more or less the same as that in which the im
portance of this minimum decreases. At the same time — and
this is important — the order is such that the first places are
very near some point to the west of Scotland, and that the
others gradually go further and further away from that centre.
Onlv four of these show a decided minimum : the most
pronounced one occurs at Arbroath in the east of Scotland,
and the next at Valencia ; the two others are Rothesay near
the west coast of Scotland and Thorshavn on the Faroe Islands.
After these places the minimum assumes a less decided
character ; it is no longer a valley, but only a less steep
incline of the ascending curve. These stations form another
circle round Scotland ; they are : Berufjord. on the east coast
Range of Meteorological and Magnetic Phenomena. 63
of Iceland*, Christiansand, on the west coast of Norway,
Greenwich, and the stations in the Netherlands.
Much importance must be attached to the fact that at
Stykkisholm, on the west coast of Iceland, this minimum is
not at all perceptible, and hardly, if at all, at Brest. And
some importance also to the fact that it sets in first at the two
Scotch stations, next at Valencia and the Faroer.
If you consider these facts I think that you will own that
here is an anomaly which must have its origin at the coast of
the west of Scotland, and probably at no great distance. For
if this origin were at a greater distance from the Scottish
coast, it would be hardly conceivable that its influence should
die out so rapidly on the two straight lines Thorshavn,
Berufjord, Stykkisholm, and Scotland, England, Brest
Thus far 1 believe my conjecture is backed by what con
stitutes a certain amount of proof. But the road, or rather
roads, which this minimum subsequently follows through
part of Europe is decidedly bewildering. Brussels shows it,
Paris most decidedly; so do Lyons, Montpellier, Triest, and
Klagonfurt; while, curiously enough, south and north of the
line uniting these stations, both in the German and the
Italian stations, I cannot find a trace of it.
But in the North of Europe also there are decidedly some
traces of it. Both Copenhagen and Haparanda show them,
so do Baltischport and Kem on the western side of the
White Sea. In this part of Europe the boundary beyond
which this feature is not traceable is a line passing w r est of
Konigsberg, Petersburg, and Archangel. Some traces of
this minimum are also to be found in a group of stations of
which Warsaw is the northernmost.
There is another fact which may be connected with the
strange distribution of this minimum. Nearly in the whole
of Russia and the North of Europe the highest point of the
whole curve has a strong tendency to occur in August, in
the very beginning for the eastern stations with a single
maximum, while those with a double summer maximum show
the highest one about the middle of August. This may,
for instance, be seen in the curves for Thorshavn, Berufjord,
and Christiansand. In the west and south of Europe, on the
contrary, the July maximum, on an average, is decidedly,
* I owe these northern stations to the kindness of Dr. A. Paulsen,
Director of the Meteorological Observatory at Copenhagen. I owe thanks
to a great many more gentlemen for the kindness with which material,
even manuscript, was placed at my disposal ; but their names will come
more naturally when the time shall ultimately have come to publish the
whole of this investigation.
64 Dr. van Rijckevorsel on Irregularities in the Yearly
although not much, higher than the August one. This is an
indication of the southern origin of this July maximum. May
not this be an explanation ? Lnte in June Europe is invaded
from a point off the Scottish coast by a strong minimum, but
a week or so later by a strong maximum from the south or
southwest, let us say tropical Africa. Therefore it is probable
that the fight, so to speak, of these two features, in the first
place causes the distribution just spoken of, and that the
curious particulars w r hich accompany it must find their expla
nation in local circumstances.
A wild hypothesis ? I know it is. But 1 give it in order
to show how vividly some meteorological problems are put
before us by this method, and how a path is shown at the
same time to their evolution. For can there be much doubt
but that, as soon as we shall be possessed of more data, not
about temperature only, more and more of these problems must
make great progress towards a solution ? One curve for
magnetic declination or for rainfall may throw a ray of light
on an intricate point which at once solves it.
Another argument to the same purpose. There will be
seen, not without some effort perhaps, a very slight minimum
between the 10th and 15th of May in the three upper curves
(PI. II.) . There are other features of the same importance, of a
greater one evem There need be no fear of my discussing them
all, for the very plausible reason that I do not know anything
about them. They may either prove interesting some day,
or vanish completely, when meteorologists shall be able to
discuss longer series of observations. But this one is worth a
moment's notice, for it is really the mark of the somuch
discussed u Icesaints." You will notice a trace of it in the
three upper curves, not in the two lower ones. (The rainfall
is too doubtful to be quoted here.) Well, these two lower
curves belong to Greenwich, and you will see that the hand
mark, if I may express myself thus, of these cold saints
tends to vanish in the British Isles. The temperaturecurve
for Greenwich indeed shows it still, that for Valencia perhaps,
but certainly in Scotland it is not visible. Here again the
southern, or southeastern, origin of this phenomenon, which I
hinted at at Toronto, is confirmed by other curves than those
for the temperature.
As hinted at before, some of these curves leave for the
moment a doubt. It will be seen that between July and
November I left in some of the vertical lines a break in the lower
regions of my diagram. In nearly every case it is only the
barometer that is at fault. I already have stated the cause:
the series of 19 years which I used is not long enough. I
regret that 1 have been unable to be ready in time with a
Range of Meteorological and Magnetic Phenomena. 65
longer series. But I fully expect that when this is quite
computed these doubts will be cleared away.
There is, however, another class of doubts which must
perhaps be allowed as yet to stand over. It will be seen that
between the maxima marked 15 and 16 I suggest for the
upper curves two minima and an unimportant maximum
between them, while lower down I have only been able to
trace a single minimum. This may of course be consistent
with the facts. Eventually for one phenomenon a certain
maximum may become so slight that it can no longer be
detected, at least not by our present methods ; and thus the
two minima before and after it may merge into one, although
other phenomena show two minima.
Another fact of the same nature occurs perhaps between the
July and August maxima, between the vertical lines 9 and 10.
The dotted line between these two is much more crooked than
the lines 9 and 10 seem to warrant. On the other hand,
nearly all the lines show very slight indications that this
minimum is indeed a double one separated by a very slight
maximum. But this feature is so faint that I did not feel jus
tified for the present in drawing the vertical lines accordingly.
Again the line 17 is a doubtful one. Of course this is
partly only a technical difficulty. Where the curves are so
near together as in this part of the diagram, the vertical lines
must seem — not really be — more crooked than they are in the
summer part of the diagram, where they expand to such a
length. Still I have a feeling that line 17 is also in reality
composed of two maxima, of which one, in December, ought
to have the number 17, and another, chiefly in January, the
new number 18.
I might of course multiply instances like these, but I feel
that this is at present useless. The seventeen lines which I
have numbered will, I think, be found in future to be a near
approximation to truth for Western Europe so far as the most
remarkable features go. But numberless other subsidiary
vertical lines may eventually be filled in in course of time
when more and better material shall be at hand than I could
avail myself of as yet.
But this, I think, is the result of this part of the work :
There is one potent cause which for a large part rules all
meteorological and magnetic phenomena, and influences them
all in a similar way nearly simultaneously.
1 do not of course pretend to teach a new doctrine in these
words, but I think I have shown a new proof of it. And
also how largely the simple method which 1 have used may in
future contribute to the solution of meteorological problems.
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. F
66 Lord Kelvin on the Age of the
V. The Age of the Earth as an Abode fitted for Life.
By the Eight Hon. Lord Kelvin, G.C.V.O. *
§ 1. rilHE age of the earth as an abode fitted for life is cer
JL tainly a subject which largely interests mankind in
general. For geology it is of vital and fundamental impor
tance — as important as the date of the battle of Hastings is
for English history — yet it was very little thought of by
geologists of thirty or forty years ago ; how little is illus
trated by a statement f, which I will now read, given originally
from the presidential chair of the Geological Society by
Professor Huxley in 1869, when for a second time, after a
seven years' interval, he was president of the Society.
" I do not suppose that at the present day any geologist would he found
... to deny that the rapidity of the rotation of the earth may be
diminishing, that the sun may be waxing dim, or that the earth itself may
be cooling. Most of us, I suspect, are Gallios, ' who care for none of
these things/ being of opinion that, true or fictitious, they have made no
practical difference to the earth, during the period of which a record is
preserved in stratified deposits."
§ 2. I believe the explanation of how it was possible for
Professor Huxley to say that he and other geologists did not
care for things on which the age of life on the earth essen
tially depends, is because he did not know that there was
valid foundation for any estimates worth considering as to
absolute magnitudes. If science did not allow us to give any
estimate whatever as to whether 10,000,000 or 10,000,000,000
years is the age of this earth as an abode fitted for life, then
1 think Professor Huxley would have been perfectly right in
saying that geologists should not trouble themselves about
it, and biologists should go on in their own way, not en
quiring into things utterly beyond the power of human
understanding and scientific investigation. This would have
left geology much in the same position as that in which
English history would be if it were impossible to ascertain
w T hether the battle of Hastings took place 800 years ago, or
800 thousand years ago, or 800 million years ago. If it were
absolutely impossible to find out which of these periods is
more probable than the other, then I agree we might be
Gallios as to the date of the Norman Conquest. But a
* Communicated by the Author, being the 1897 Annual Address of the
Victoria Institute with additions written at different times from June
1897 to May 1898.
t In the printed quotations the italics are mine in every case, not so
the capitals in the quotation from Page's Textbook,
Earth as an Abode fitted for Life. 67
change took place just about the time to which I refer, and
from then till now geologists have not considered the question
of absolute dates in their science as outside the scope of their
investigations.
§ 3. I may be allowed to read a few extracts to indicate
how geological thought was expressed in respect to this
subject, in various largely used popular textbooks, and in
scientific writings which were new in 1868, or not so old as
to be forgotten. I have several short extracts to read and
I hope you will not find them tedious.
The first is three lines from Darwin's " Origin of Species,"
1859 Edition, p. 287.
" In all probability a far longer period than 300,000,000 years has
elapsed since the latter part of tlie secondary period."
Here is another still more important sentence, which I read
to you from the same book : —
" He who can read Sir Charles Lyell's grand work on the Principles
of Geology, which the future historian will recognize as having produced
a revolution in natural science, yet does not admit how incomprehensibly
vast have been the past periods of time, may at once close this volume"
1 shall next read a short statement from Page's ' Advanced
Students' TextBook of Geology,' published in 1859 : —
" Again where the FORCE seems unequal to the result, the student
should never lose sight of the element TIME : an element to ivhich we can
set no bounds in the past, any more than we know of its limit in the
future." _
" It will be seen from this hasty indication that there are two great
schools of geological causation the one ascribing every result to the
ordinary operations of Nature, combined with the element of unlimited
time, the other appealing to agents that operated during the earlier
epochs of the world with greater intensity, and also for the most part
over wider areas. The former belief is certainly more in accordance with
the spirit of right philosophy, though it must be confessed that many
problems in geology seem to find their solution only through the
admission of the latter hypothesis."
§ 4. I have several other statements which I think you may
hear with some interest. Dr. Samuel Haughton, of Trinity
College, Dublin, in his ' Manual of Geology/ published in
1865, p. 82, says :• —
"The infinite time of the geologists is in the past ; and most of their
speculations regarding this subject seem to imply the absolute infinity of
time, as if the human imagination was unable to grasp the period of time
requisite for the formation of a few inches of sand or feet of mud, and its
subsequent consolidation into rock." (This delicate satire is certainly
not overstrained.)
" Professor Thomson has made an attempt to calculate the length of
time during which the sun can have gone on burning at the present rate,
and has come to the following conclusion : — ' It seems, on the whole, most
¥2
8 Lord Kelvin on the Age of the
probable that the sun has not illuminated the earth for 100,000,000 years,
and almost certain that he has not done so for 500,000,000 years. As for
the future, we may say with equal certainty, that the inhabitants of the
earth cannot continue to enjoy the light and heat essential to their life
for many million years longer, unless new sources, now unknown to us,
are prepared in the great storehouse of creation."
I said that in the sixties and I repeat it now ; but with
charming logic it is held to be inconsistent with a later state
ment that the sun has not been shining 60,000,000 years ;
and that both that and this are stultified by a still closer
estimate which says that probably the sun has not been
shining for 30,000,000 years ! And so my efforts to find some
limit or estimate for Geological Time have been referred to
and put before the public, even in London daily and weekly
papers, to show how exceedingly wild are the wanderings
of physicists, and how mutually contradictory are their con
clusions, as to the length of time which has actually passed
since the early geological epochs to the present date.
Dr. Haughton further goes on —
" This result (100 to 500 million years) of Professor Thomson's, although
very liberal in the allowance of time, has offended geologists, because, having
been accustomed to deal with time as an infinite quantity at their disptosal,
they feel naturally embarrassment and alarm at any attempt of the science
of Physics to place a limit upon their speculations. It is quite possible
that even a hundred million of years may be greatly in excess of the
actual time during which the sun's heat has remained constant."
§ 5. Dr. Haughton admitted so much with a candid open
mind; but he went on to express his own belief (in 1865) thus: —
" Although I have spoken somewhat disrespectfully of the geological
calculus in my lecture, yet I believe that the time during which organic
life has existed on the earth is practically infinite, because it can be shown
to be so great as to be inconceivable by beings of our limited intelligence."
Where is inconceivableness in 10,000,000,000 ? There is
nothing inconceivable in the number of persons in this room,
or in London. We get up to millions quickly. Is there any
thing inconceivable in 30,000,000 as the population of
England, or in 38,000,000 as the population of Great Britain
and Ireland, or in 352,704,863 as the population of the
British Empire ? Not at all. It is just as conceivable as half
a million years or 500 millions.
§ 6. The following statement is from Professor Jukes's
' Students' Manual of Geology : ' —
" The time required for such a slow process to effect such enormous
results must of course be taken to be inconceivably great. The word
' inconceivably ' is not here used in a vague but in a literal sense, to
indicate that the lapse of time required for the denudation that has
produced the present surfaces of some of the older rocks, is vast beyond
any idea of time which the human mind is capable of conceiving."
Earth as an Abode fitted for Life. 69
u Mr. Darwin, in Ms admirably reasoned book on the origin of species,
so full of information and suggestion on all geological subjects, estimates
the time required for the denudation of the rocks of the Weald of Kent,
or the erosion of spac3 between the ranges of chalk hills, known as the
North and South Downs, at three hundred millions of years. The
grounds for forming this estimate are of course of the vaguest de
scription. It may be possible, perhaps, that the estimate is a hundred
times too great, and that the real time elapsed did not exceed three
million years, but, on the other hand, it is just as likely that the time
which actually elapsed since the first commencement of the erosion till it
was nearly as complete as it now is, was really a hundred times greater
than his estimate, or thirty thousand millions of years."
§ 7. Thus Jukes allowed estimates of anything from
3 millions to 30,000 millions as the time which actually
passed during the denudation of the Weald. On the other
hand Professor Phillips in his Rede lecture to the University
of Cambridge (18(30), decidedly prefers one inch per annum
to Darwin's one inch per century as the rate of erosion : and
says that most observers would consider even the one inch
per annum too small for all but the most invincible coasts !
He thus, on purely geological grounds, reduces Darwin's
estimate of the time to less than one onehundredth. And,
reckoning the actual thicknesses of all the known geological
strata of the earth, he finds 96 million years as a possible
estimate for the antiquity of the base of the stratified rocks ;
but he gives reasons for supposing that this may be an over
estimate, and he finds that from stratigraphical evidence
alone, we may regard the antiquity of life on the earth as
possibly between 38 millions and 96 millions of years.
Quite lately a very careful estimate of the antiquity of
strata containing remains of life on the earth has been
given by Professor Sollas, of Oxford, calculated according
to stratigraphical principles which had been pointed out by
Mr. Alfred Wallace. Here it is * : — " So far as I can at
present see, the lapse of time since the beginning of the
Cambrian system is probably less than 17,000,000 years,
even when computed on an assumption of uniformity, which
to me seems contradicted by the most salient facts of geology.
Whatever additional time the calculations made on physical
data can afford us, may go to the account of preCambrian
deposits, of which at present we know too little to serve for
an independent estimate."
§ 8. In one of the evening Conversaziones of the British
Association during its meeting at Dundee in 1867 I had a
conversation on geological time with the late Sir Andrew
Ramsay, almost every word of which remains stamped on
* "The Age of the Earth," < Nature,' April 4th, 1895.
70 Lord Kelvin on the Age of the
my mind to this day. We had been hearing a brilliant and
suggestive lecture by Professor (now Sir Archibald) Geikie
on the geological history of the actions by which the existing
scenery of Scotland was produced. I asked Ramsay how long
a time he allowed for that history. He answered that he
could suggest no limit to it. I said, " You don't suppose
things have been going on always as they are now ? You
don't suppose geological historvhas run through 1,000,000,000
years?" "Certainly I do/' " 10,000,000,000 years V } ' ; Yes."
" The sun is a finite body. You can tell how many tons it is.
Do you think it has been shining on for a million million
years ? " "I am as incapable of estimating and under
standing the reasons which you physicists have for limiting
geological time as you are incapable of understanding the
geological reasons for our unlimited estimates." I answered,
" You can understand physicists' reasoning perfectly if you
give your mind to it." I ventured also to say that physicists
were not wholly incapable of appreciating geological diffi
culties ; and so the matter ended, and we had a friendly
agreement to temporarily differ.
§ 9. In fact, from about the beginning of the century till
that time (1867), geologists had been nurtured in a philosophy
originating with the Huttonian system: much of it substantially
very good philosophy, but some of it essentially unsound and
misleading : witness this, from Playfair, the eloquent and able
expounder of Hutton : —
"How often these vicissitudes of decay arid renovation have been
repeated is not for us to determine ; they constitute a series of which as
the author of this theory has remarked, we neither see the beginning nor
the end ; a circumstance that accords well with what is known concerning
other parts of the economy of the world. In the continuation of the
different species of animals and vegetables that inhabit the earth, we
discern neither a beginning nor an end : in the planetary motions where
geometry has carried the eye so far both into the future 'and the past we
discover no mark either of the commencement or the termination of the
present order."
§ 10. Led by Hutton and Playfair, Lyell taught the
doctrine of eternity and uniformity in geologv : and to
explain plutonic action and underground heat, invented a
thermoelectric " perpetual " motion on which, in the vear
1862, in my paper on the " Secular Cooling of the Earth"*
published in the " Transactions of the Royal Societv of Edin
burgh/ I commented as follows : —
* Reprinted in Thomson and Tait, ' Treatise on Natural Philosophy,'
1st and 2nd Editions, Appendix D (g).
Earth as an Abode fitted for Life. 71
"To suppose, as Lyell, adopting the chemical hypothesis, has done*,
that the substances, combining together, may be again separated electro
lytically by thermoelectric currents, due to the heat generated by their
combination, and thus the chemical action and its heat continued in an
endless cycle, violates the principles of natural philosophy in exactly the
same manner, and to the same degree, as to believe that a clock con
structed with a selfwinding movement may fulfil the expectations of its
ingenious inventor by going for ever."
It was only by sheer force of reason that geologists have
been compelled to think otherwise, and to see that there was
a definite beginning, and to look forward to a definite end,
of this world as an abode fitted for life.
§ 11. It is curious that English philosophers and writers
should not have noticed how Newton treated the astro
nomical problem. Play fair, in what I have read to yon, speaks
of the planetary^ system as being absolutely eternal, and
unchangeable : having had no beginning and showing no
signs of progress towards an end. He assumes also that the
sun is to go on shining for ever, and that the earth is to go on
revolving round it for ever. He quite overlooked Laplace's
nebular theory ; and he overlooked Newton's counterblast
to the planetary " perpetual motion.'" Newton, commenting
on his own f First Law of Motion,' says, in his terse Latin,
which I will endeavour to translate, a But the greater bodies
of planets and comets moving in spaces less resisting, keep
their motions longer," That is a strong counterblast against
any idea of eternity in the planetary system.
§ 12. I shall now, without further preface, explain, and I
hope briefly, so as not to wear out your patience, some of the
arguments that I brought forward between 1862 and 1869,
to show strict limitations to the possible age of the earth as
an abode fitted for life.
Kant f pointed out in the middle of last century, what had
not previously been discovered by mathematicians or physical
astronomers, that the frictional resistance against tidal cur
rents on the earth's surface must cause a diminution of the
earth's rotational speed. This really great discovery in
* 'Principles of Geology,' chap. xxxi. ed. 1853.
f In an essay first published in the Koenigsberg Nackrichten, 1754,
Nos. 23, 24 ; having been written with reference to the offer of a prize by
the Berlin Academy of Sciences in 1754. Here is the titlepage, in full,
as it appears in vol. vi. of Kant's Collected Works, Leipzig, 1839 : —
Untersuchung der Frage : Ob die Erde in ihrer Umdrenung um die Achse,
wodurch sie die Abwechselung des Tages und der Nacht hervorbringt,
einige Veranderung seit den ersten Zeiten ihres Ursprunges erlitten habe,
welches die Ursache davon sei, und woraus man sich ihrer versichern
konne ? welche von der Koniglichen Akademie der Wissenschaften zu
Berlin zum Preise aufgegeben worden, 1754.
72 Lord Kelvin on the Age of the
Natural Philosophy seems to have attracted very little
attention, — indeed to have passed quite unnoticed, — among
mathematicians, and astronomers, and naturalists, until about
1840, when the doctrine of energy began to be taken to
heart. In 1866, Delaunay suggested that tidal retardation
of the earth's rotation was probably the cause of an out
standing acceleration of the moon's mean motion reckoned
according to the earth's rotation as a timekeeper found by
Adams in 1853 by correcting a calculation of Laplace which
had seemed to prove the earth's rotational speed to be uni
form *. Adopting Delaunay's suggestion as true, Adams, in
conjunction with Professor Tait and myself, estimated the
diminution of the earth's rotational speed to be such that
the earth as a timekeeper, in the course of a century, would
get 22 seconds behind a thoroughly perfect watch or clock
rated to agree with it at the beginning of the century.
According to this rate of retardation the earth, 7,200 million
years ago, would have been rotating twice as fast as now :
and the centrifugal force in the equatorial regions would
have been four times as great as its present amount, which
is 2J9 of gravity. At present the radius of the equatorial
sealevel exceeds the polar semidiameter by 21J kilometres,
which is, as nearly as the most careful calculations in the
theory of the earth's figure can tell us, just what the excess
of equatorial radius of the surface of the sea all round would
be if the whole material of the earth were at present liquid
and in equilibrium under the influence of gravity and centri
fugal force with the present rotational speed, and £ of what
it would be if the rotational speed were twice as great.
Hence, if the rotational speed had been twice as great as
its present amount when consolidation from approximately
the figure of fluid equilibrium took place, and if the solid
earth, remaining absolutely rigid, had been gradually slowed
down in the course of millions of years to its present speed
of rotation, the water would have settled into two circular
oceans round the two poles : and the equator, dry all round,
would be 64*5 kilometres above the level of the polar sea
bottoms. This is on the supposition of absolute rigidity of
the earth after primitive consolidation. There would, in
reality, have been some degree of yielding to the gravitational
tendency to level the great gentle slope up from each pole to
equator. But if the earth, at the time of primitive consolida
* ' Treatise on Natural Philosophy ' (Thomson and Tait), §830, ed. 1,
1867, and later editions ; also ' Popular Lectures and Addresses/ vol. ii.
(Kelvin), ' Geological Time,' being a reprint of an article communicated
to the Glasgow Geological Society, February 27th, 1868.
Earth as an Abode fitted for Life. 73
tion, had bean rotating twice as fast as at present, or even
20 per cent, faster than at present, traces of it's present figure
must have been left in a great preponderance of land, and
probably no sea at all, in the equatorial regions. Taking
into account all uncertainties, whether in respect to Adams'
estimate of the rate of frictional retardation of the earth's
rotatory speed, or to the conditions as to the rigidity of the
earth once consolidated, we may safely conclude that the
earth was certainly not solid 5,000 million years ago, and
was probably not solid 1,000 million years ago*.
§ 13. A second argument for limitation of the earth's age,
which was really my own first argument, is founded on the
consideration of underground heat. To explain a first rough
and ready estimate of it I shall read one short statement.
It is from a very short paper that I communicated to the
Eoyal Society of Edinburgh on the 18th December, 1865,
entitled, " The Doctrine of Uniformity in Geology brief! y
refuted."
" The ' Doctrine of Uniformity ' in Geology, as held by many of the
most eminent of British Geologists, assumes that the earth's surface and
upper crust have been nearly as they are at present in temperature, and
other physical qualities, during millions of millions of years. But the heat
which we knoio, by observation, to be now conducted out of the earth yearly
is so great, that if this action had been going on with any approach to
uniformity for 20,000 million years, the amount of heat lost out of the
earth would have been about as much as would heat, by 100° C, a
quantity of ordinary surface rock of 100 times the earth's bulk. This
would be more than enough to melt a mass of surface rock equal in bulk
to the whole earth. No hypothesis as to chemical action, internal fluidity,
effects of pressure at great depth, or possible character of substances in
the interior of the earth, possessing the smallest vestige of probability,
can justify the supposition that the earth's upper crust has remained
nearly as it is, while from the whole, or from any part, of the earth, so
great a quantity of heat has been lost."
§ 14. The sixteen words which I have emphasized in read
ing this statement to you (italics in the reprint) indicate the
matteroffact foundation for the conclusion asserted. This
conclusion suffices to sweep away the whole system of geolo
gical and biological speculation demanding an " inconceiv
ably " great vista of past time, or even a few thousand million
years, for the history of life on the earth, and approximate
uniformity of plutonic action throughout that time ; which,
as we have seen, was very generally prevalent thirty years
* " The fact that the continents are arranged along meridians, rather
than in an equatorial belt, affords some degree of proof that the consoli
dation of the earth took place at a time when the diurnal rotation differed
but little from its present value. It is probable that the date of consoli
dation is considerably more recent than a thousand million years ago." —
Thomson and Tait, 'Treatise on Natural Philosophy,' 2nd ed., 1883, § 830.
74 Lord Kelvin on the Age of the
ago among British Geologists and Biologists; and which, I
must say, some of our chiefs of the present day have not yet
abandoned. Witness the Presidents of the Geological and
Zoological Sections of the British Association at its meetings
of 1893 (Nottingham), and of 1896 (Liverpool).
Mr. Teall : Presidential Address to the Geological Section, 1 893.
" The good old British ship ' Uniformity,' built by Hutton and refitted
by Lyell, has won so mauy glorious victories in the past, and appears still
to be in such excellent righting trim, that I see no reason why she should
haul down her colours either to ' Catastrophe' or ' Evolution.' Instead,
therefore, of acceding to the request to ' hurry up ' we make a demand
for more time."
Professor Poulton : Presidential Address to the Zoological Section, 1896.
" Our argument does not deal with the time required for the origin of
life, or for the development of the lowest beings with which we are
acquainted from the first formed beings, of which we know nothing.
Both these processes may have required an immensity of time ; but as we
know nothing whatever about them and have as yet no prospect of
acquiring any information, we are compelled to confine ourselves to as
much of the process of evolution as we can infer from the structure of
living and fossil forms — that is, as regards animals, to the development
of the simplest into the most complex Protozoa, the evolution of the
Metazoa from the Protozoa, aud the branching of the former into its
numerous Phyla, with all their Classes, Orders, Families, Genera, and
Species. But we shall find that this is quite enough to necessitate a very
large increase in the time estimated by the geologist."
§ 15. In my own short paper from which I have read you
a sentence, the rate at which heat is at the present time lost
from the earth by conduction outwards through the upper
crust, as proved by observations of underground temperature
in different parts of the world, and by measurement of the
thermal conductivity of surface rocks and strata, sufficed to
utterly refute the Doctrine of Uniformity as taught by Hutton,
Lyell, and their followers ; which was the sole object of that
paper.
§ 16. In an earlier communication to the Royal Society
of Edinburgh *, I had considered the cooling of the earth
due to this loss of heat ; and by tracing backwards the
process of cooling had formed a definite estimate of the
greatest and least number of million years which can
possibly have passed since the surface of the earth was
everywhere red hot. I expressed my conclusion in the
following statement f : —
* " On the Secular Cooling of the Earth," Trans. Roy. Soc. Edinburgh,
vol. xxiii. April 28th, 1862, reprinted in Thomson and Tait, vol. iii.
pp. 468485, and Math, and Phys. Papers, art. xciv. pp, 295311.
t " On the Secular Cooling of the Earth," Math, and Phys. Papers,
vol. iii. § 11 of art. xciv.
Earth as an Abode fitted for Life. 75
" We are very ignorant as to the effects of high, temperatures in altering
the conductivities and specific heats and melting temperatures of rocks,
and as to their latent heat of fusion. We must, therefore, allow very
wide limits in such an estimate as I have attempted to make ; but I think
we may with much probability sa} r that the consolidation cannot have
taken place less than 20 million years ago, or we should now have more
underground heat than we actually have ; nor more than 400 million years
ago, or we should now have less underground heat than we actually have.
That is to say, I conclude that Leibnitz's epoch of emergence of the
consistentior status [the consolidation of the earth from red hot or white
hot molten matter] was probably between those dates."
§ 17. During the 35 years which have passed since I gave
this wideranged estimate, experimental investigation has
supplied much of the knowledge then wanting regarding the
thermal properties of rocks to form a closer estimate of the
time which has passed since the consolidation of the earth,
and we have now good reason for judging that it was more
than 20 and less than 40 million years ago ; and probably
much nearer 20 than 40,
§ 18. Twelve years ago, in a laboratory established by
Mr. Clarence King in connexion with the United States
Geological Survey, a very important series of experimental
researches on the physical properties of rocks at high
temperatures was commenced by Dr. Carl Barus, for the
purpose of supplying trustworthy data for geological theory.
Mr, Clarence King, in an article published in the 'American
Journal of Science ; *, used data thus supplied, to estimate
the age of the earth more definitely than was possible for me
to do in 1862, with the very meagre information then available
as to the specific heats, thermal conductivities, and tempera
tures of fusion, of rocks. I had taken 7000° F. (3781° C.)
as a high estimate of the temperature of melting rock. Even
then I might have taken something between 1000° C. and
2000° C. as more probable, but I was most anxious not to
underestimate the age of the earth, and so I founded my
primary calculation on the 7000° F. for the temperature of
melting rock. We know now from the experiments of Carl
Barus f that diabase, a typical basalt of very primitive
character, melts between 1100° C. and 1170°, and is tho
roughly liquid at 1200°. The correction from 3871° C. to
1200° or 1/3*22 of that value, for the temperature of solidifi
cation, would, with no other change of assumptions, reduce
my estimate of 100 million to 1/(3*22) 2 of its amount, or a
little less than 10 million years ; but the effect of pressure
on the temperature of solidification must also be taken into
* * On the Age of the Earth,' vol. xlv. January 1893.
t Phil. Mag. 1893, first halfyear, pp. 186, 187, 301305.
76 Lord Kelvin on the Age of the
account, and Mr. Clarence King, after a careful scrutiny of
all the data given him for this purpose by Dr. Barus, concludes
that without further experimental data " we have no warrant
for extending the earth's age beyond 24 millions of years."
§ 19. By an elaborate piece of mathematical bookkeeping
1 have worked out the problem of the conduction of heat
outwards from the earth, with specific heat increasing up to
the meltingpoint as found by Riicker and RobertsAusten
and by Barus, but with the conductivity assumed constant ;
and, by taking into account the augmentation of melting
temperature with pressure in a somewhat more complete
manner than that adopted by Mr. Clarence King, I am not
led to differ much from his estimate of 24 million years.
But, until we know something more than we know at present
as to the probable diminution of thermal conductivity with
increasing temperature, which would shorten the time since
consolidation, it would be quite inadvisable to publish any
closer estimate.
§ 20. All these reckonings of the history of underground
heat, the details of which I am sure you do not wish me to
put before you at present, are founded on the very sure
assumption that the material of our present solid earth all
round its surface was at one time a white hot liquid. The
earth is at present losing heat from its surface all round from
year to year and century to century. We may dismiss as
utterly untenable any supposition such as that a few thousand
or a few million years of the present regime in this respect
was preceded by a few thousand or a few million years of
heating from without. History, guided by science, is bound
to find, if possible, an antecedent condition preceding every
known state of affairs, whether of dead matter or of living
creatures. Unless the earth was created solid and hot out
of nothing, the regime of continued loss of heat must have
been preceded by molten matter all round the surface.
§ 21. 1 have given strong reasons * for believing that
immediately before solidification at the surface, the interior
was solid close up to the surface : except comparatively small
portions of lava or melted rock among the solid masses of
denser solid rock which had sunk through the liquid, and
possibly a somewhat large space around the centre occupied
by platinum, gold, silver, lead, copper, iron, and other dense
metals, still remaining liquid under very high pressure.
§ 22. I wish now to speak to you of depths below the
* « On the Secular Cooling of the Earth," vol. iii. Math, and Phys.
Papers, §§ 1933.
Earth as an Abode fitted for Life. 11
great surface of liquid lava bounding the earth before
consolidation ; and of mountain heights and ocean depths
formed probably a few years after a first emergence of solid
rock from the liquid surface (see § 24, below) , which must have
been quickly followed by complete consolidation all round the
globe. But I must first ask you to excuse my giving you all
my depths, heights, and distances, in terms of the kilometre,
being about sixtenths of that very inconvenient measure the
English statute mile, which, with all the other monstrosities
of our British metrical system, will, let us hope, not long
survive the legislation of our present Parliamentary session
destined to honour the sixty years'' Jubilee of Queen Victoria's
reign by legalising the French metrical system for the United
Kingdom.
§ 23. To prepare for considering consolidation at the surface
let us go back to a time (probably not more than twenty
years earlier as we shall presently see — § 24) when the solid
nucleus was covered with liquid lava to a depth of several
kilometres ; to fix our ideas let us say 40 kilometres (or 4
million centimetres). At this depth in lava, if of specific
gravity 2*5, the hydrostatic pressure is 10 tons weight (10
million grammes) per square centimetre, or ten thousand
atmospheres approximately. According to the laboratory
experiments of Clarence King and Carl Barus * on Diabase,
and the thermodynamic theory f of my brother, the late
Professor James Thomson, the melting temperature of
diabase is 1170° C. at ordinary atmospheric pressure, and
would be 1420° under the pressure of ten thousand atmo
spheres, if the rise of temperature with pressure followed the
law of simple proportion up to so high a pressure.
§ 24. The temperature of our 40 kilometres deep lava
ocean of melted diabase may therefore be taken as but
little less than 1420° from surface to bottom. Its surface
would radiate heat out into space at some such rate as two
(grammewater) thermal units Centigrade per square centi
metre per second J. Thus, in a year (31J million seconds)
* Phil. Mag. 1893, first halfyear, p. 306.
t Trans. Roy. Soc. Edinburgh, Jan. 2, 1849 ; Cambridge and Dublin
Mathematical Journal, Nov. 1850. Reprinted in Math, and Phys. Papers
(Kelvin), vol. i. p. 156.
% This is a very rough estimate which I have formed from consideration
of J. T. Bottomley's accurate determinations in absolute measure of
thermal radiation at temperatures up to 9^0° C. from platinum wire and
from polished and blackened surfaces of various hinds in receivers of air
pumps exhausted down to one tenmillionth of the atmospheric pressure
Phil, Trans. Roy. Soc, 1887 and 1893.
78 Lord Kelvin on the Age of the
63 million thermal units would be lost per square centimetre
from the surface. This is, according to Carl Barus, very
nearly equal to the latent heat of fusion abandoned by a
million cubic centimetres of melted diabase in solidifying into
the glassy condition (pitchstone) which is assumed when
the freezing takes place in the course of a few minutes.
But, as found by Sir James Hall in his Edinburgh experi
ments *" of 100 years ago, when more than a few minutes is
taken for the freezing, the solid formed is not a glass but a
heterogeneous crystalline solid of rough fracture ; and if a
few hours or days, or any longer time, is taken, the solid
formed has the well known rough crystalline structure of
basaltic rocks found in all parts of the world. Now Carl
Barus finds that basaltic diabase is 14 per cent, denser than
melted diabase, and 10 per cent, denser than the glass pro
duced by quick freezing of the liquid. He gives no data,
nor do Riicker and RobertsAusten, who have also experi
mented on the thermodynamic properties of melted basalt,
give any data, as to the latent heat evolved in the consolida
tion of liquid lava into rock of basaltic quality. Guessing
it as three times the latent heat of fusion of the diabase
pitch stone, I estimate a million cubic centimetres of liquid
frozen per square centimetre per centimetre per three years.
This would diminish the depth of the liquid at the rate of a
million centimetres per three years, or 40 kilometres in twelve
years.
§ 25. Let us now consider in what manner this diminution
of depth of the lava ocean must have proceeded, by the
freezing of portions of it ; all having been at temperatures
very little below the assumed 1420° melting temperature of
the bottom, when the depth was 40 kilometreSo The loss of
heat from the whitehot surface (temperatures from 1420° to
perhaps 1380° in different parts) at our assumed rate of two
(grammewater Centigrade) thermal units per sq. cm. per
sec. produces very rapid cooling of the liquid within a few
centimetres of the surface (thermal capacity *36 per gramme,
according to Barus) and in consequence great downward
rushes of this cooled liquid, and upwards of hot liquid,
spreading out horizontally in all directions when it reaches
the surface. When the sinking liquid gets within perhaps
20 or 10 or 5 kilometres of the bottom, its temperature f
* Trans. Roy. Soc. Edinburgh.
t Tlie temperature of the sinking liquid rock rises in virtue of the
increasing pressure : but much less than does the freezing point of the
liquid or of some of its ingredients. (See Kelvin, Math. andPhys. Papers,
vol. iii. pp. 69, 70.)
Earth as an Abode fitted for Life. 79
becomes the freezingpoint as raised by the increased
pressure ; or, perhaps more correctly stated, a temperature
at which some of its ingredients crystallize out of it. Hence,
beginning a few kilometres above the bottom, we have a
snow shower of solidified lava or of crystalline flakes, or
prisms, or granules of felspar, mica, hornblende, quartz, and
other ingredients : each little crystal gaining mass and falling
somewhat faster than the descending liquid around it, till it
reaches the bottom. This process goes on until, by the
heaping of granules and crystals on the bottom, our lava
ocean becomes silted up to the surface.
Probable Origin of Granite. (§§ 26, 27.)
§ 26. Upon the suppositions we have hitherto made we
have, at the stage now reached, all round the earth at the
same time a red hot or white hot surface of solid granules or
crystals with interstices filled by the mother liquor still
liquid, but ready to freeze with the slightest cooling. The
thermal conductivity of this heterogeneous mass, even
before the freezing of the liquid part, is probably nearly
the same as that of ordinary solid granite or basalt at a
red heat, which is almost certainly * somewhat less than the
thermal conductivity of igneous rocks at ordinary tempera
tures. If you wish to see for yourselves how quickly it
would cool when wholly solidified take a large macadamising
stone, and heat it red hot in an ordinary coal fire. Take it
out with a pair of tongs and leave it on the hearth, or on a
stone slab at a distance from the fire, and you will see that
in a minute or two, or perhaps in less than a minute, it cools
to below red heat.
§ 27. Half an hour f after solidification reached up to the
surface in any part of the earth, the mother liquor among the
granules must have frozen to a depth of several centimetres
below the surface and must have cemented together the
granules and crystals, and so formed a crust of primeval
granite, comparatively cool at its upper surface, and red hot
to white hot, but still all solid, a little distance down ;
becoming thicker and thicker very rapidly at first ; and
after a few weeks certainly cold enough at its outer surface
to be touched by the hand.
* Proc. K. S., May 30, 1895.
t Witness the rapid cooling of lava running red hot or white hot from
a volcano, and after a few days or weeks presenting a black hard crust
strong enough and cool enough to be walked over with impunity.
80 Lord Kelvin on the Age of the
Probable Origin of Basaltic Bock * (§§ 28, 29.)
§ 28. We have hitherto left, without much consideration,
the mother liquor among the crystalline granules at all
depths below the bottom of our shoaling lava ocean. It was
probably this interstitial mother liquor that was destined
to form the basaltic rock of future geological time. What
ever be the shapes and sizes of the solid granules when first
falling to the bottom, they must have lain in loose heaps
with a somewhat large proportion of space occupied by
liquid among them. But, at considerable distances down in
the heap, the weight of the superincumbent granules must
tend to crush corners and edges into fine powder. If the
snow shower had taken place in air we may feel pretty sure
(even with the slight knowledge which we have of the hard
nesses of the crystals of felspar, mica and hornblende, and
of the solid granules of quartz) that, at a depth of 10 kilo
metres, enough of matter from the corners and edges of the
granules of different kinds, would have been crushed into
powder of various degrees of fineness, to leave an exceed
ingly small proportionate volume of air in the interstices
between the solid fragments. But in reality the effective
weight of each solid particle, buoyed as it was by hydrostatic
pressure of a liquid less dense than itself by not more than
20 or 15 or 10 per cent., cannot have been more than from
about onefifth to onetenth of its weight in air, and there
fore the same degree of crushing effect as would have been
experienced at 10 kilometres with air in the interstices, must
have been experienced only at depths of from 50 to 100 kilo
metres below the bottom of the lava ocean.
§ 29. A result of this tremendous crushing together of the
solid granules must have been to press out the liquid from
among them, as water from a sponge, and cause it to pass
upwards through the less and less closely packed heaps of
solid particles, and out into the lava ocean above the heap.
But, on account of the great resistance against the liquid
permeating upwards 30 or 40 kilometres through interstices
among the solid granules, this process must have gone on
somewhat slowly ; and, during all the time of the shoaling of
the larva ocean, there may have been a considerable proportion
of the whole volume occupied by the mother liquor among
the solid granules, down to even as low as 50 or 100 kilo
metres below the top of the heap, or bottom of the ocean, at
* See Addendum at end of Lecture,
Earth as an Abode fitted for Life. 81
each instant. When consolidation reached the surface, the
oozing upwards of the mother liquor must have been still
going on to some degree. Thus, probably for a few years
after the first consolidation at the surface, not probably for
as long as one hundred years, the settlement of the solid
structure by mere mechanical crushing of the corners and
edges of solid granules, may have continued to cause the
oozing upwards of mother liquor to the surface through
cracks in the first formed granite crust and through fresh
cracks in basaltic crust subsequently formed above it.
Leibnitz's Consistentior Status.
§ 30. When this oozing everywhere through fine cracks
in the surface ceases, we have reached Leibnitz's consistentior
status ; beginning with the surface cool and permanently
solid and the temperature increasing to 1150° 0. at 25 or 50
or 100 metres below the surface.
Probable Origin of Continents and Ocean Depths of
the Earth. (§§ 3137.)
§ 31. If the shoaling of the lava ocean up to the surface
had taken place everywhere at the same time, the whole sur
face of the consistent solid would be the dead level of the
liquid lava all round, just before its depth became zero. On
this supposition there seems no possibility that our present
day continents could have risen to their present heights,
and that the surface of the solid in its other parts could have
sunk down to their present ocean depths, during the twenty
or twentyfive million years which may have passed since the
consistentior status began or during any time however long.
Rejecting the extremely improbable hypothesis that the conti
nents were built up of meteoric matter tossed from without,
upon the already solidified earth, we have no other possible
alternative than that they are due to heterogeneousness in
different parts of the liquid which constituted the earth before
its solidification. The hydrostatic equilibrium of the rotating
liquid involved only homogeneousness in respect to density
over every level surface (that is to say, surface perpendicular
to the resultant of gravity and centrifugal force) : it required
no homogeneousness in respect to chemical composition. Con
sidering the almost certain truth that the earth was built up of
meteorites falling together, we may follow in imagination the
whole process of shrinking from gaseous nebula to liquid lava
and metals, and solidification of liquid from central regions
outwards, without finding any thorough mixing up of dif
ferent ingredients, coming together from different directions
Phil. Mag. S. 5. Vol. 47, No. 284. Jan. 1899. G
82 Lord Kelvin on the Age of the
of space — any mixing up so thorough as to produce even
approximately chemical homogeneousness throughout every
layer of equal density. Thus we have no difficulty in under
standing how even the gaseous nebula, which at one time
constituted the matter of our present earth, had in itself a
heterogeneousness from which followed by dynamical neces
sity Europe, Asia, Africa, America, Australia, Greenland, and
the Antarctic Continent, and the Pacific, Atlantic, Indian,
and Arctic Ocean depths, as we know them at present.
§ 32. We may reasonably believe that a very slight degree
of chemical heterogeneousness could cause great differences
in the heaviness of the snow shower of granules and crystals
on different regions of the bottom of the lava ocean when
still 50 or 100 kilometres deep. Thus we can quite see how
it may have shoaled much more rapidly in some places than
in others. It is also interesting to consider that the solid
granules, falling on the bottom, may have been largely
disturbed, blown as it were into ridges (like rippled sand in
the bed of a flowing stream, or like dry sand blown into
sandhills by wind) by the eastward horizontal motion which
liquid descending in the equatorial regions must acquire,
relatively to the bottom, in virtue of the earth's rotation. It
is indeed not improbable that this influence may have been
largely effective in producing the general configuration of
the great ridges of the Andes and Rocky Mountains and of
the West Coasts of Europe and Africa. It seems, however,
certain that the main determining cause of the continents and
oceandepths was chemical differences, perhaps very slight
differences, of the material in different parts of the great lava
ocean before consolidation.
§ 33. To fix our ideas let us now suppose that over some
great areas such as those which have since become Asia,
Europe, Africa, Australia, and America, the lava ocean had
silted up to its surface, while in other parts there still were
depths ranging down to 40 kilometres at the deepest. In
a very short time, say about twelve years according to our
former estimate (§ 24) the whole lava ocean becomes silted
up to its surface.
§ 34. We have not time enough at present to think out
all the complicated actions, hydrostatic and thermodynamic,
which must accompany, and follow after, the cooling of the
lava ocean surrounding our ideal primitive continent. Bv
a hurried view, however, of the affair we see that in virtue
of, let us say, 15 per cent, shrinkage by freezing, the level
of the liquid must, at its greatest supposed depth, sink six
kilometres relatively to the continents : and thus the liquid
Earth as an Abode fitted for Life. 83
must recede from them ; and their bounding coastlines must
become enlarged. And just as water runs out of a sandbank,
drying when the sea recedes from it on a falling tide, so
rivulets of the mother liquor must run out from the edges of
the continents into the receding lava ocean. But, unlike
sandbanks of incoherent sand permeated by water remaining
liquid, our uncovered banks of whitehot solid crystals, with
interstices full of the mother liquor, will, within a few hours
of being uncovered, become crusted into hard rock by cooling
at the surface, and freezing of the liquor, at a temperature
somewhat lower than the melting temperatures of any of the
crystals previously formed. The thickness of the wholly
solidified crust grows at first with extreme rapidity, so that
in the course of three or four days it may come to be as
much as a metre. At the end of a year it may be as much
as 10 metres ; with, a surface, almost, or quite, cool enough
for some kinds of vegetation. In the course of the first few
weeks the regime of conduction of heat outwards becomes
such that the thickness of the wholly solid crust, as long as
it remains undisturbed, increases as the square root of the
time ; so that in 100 years it becomes 10 times, in 25 million
years 5000 times, as thick as it was at the end of one year :
thus, from one year to 25 million years after the time of
surface freezing, the thickness of the wholly solid crust might
grow from 10 metres to 50 kilometres. These definite num
bers are given merely as an illustration ; but it is probable
they are not enormously far from the truth in respect to what
has happened under some of the least disturbed parts of the
earth's surface.
§ 35. We have now reached the condition described above
in § 30, with only this difference, that instead of the upper
surface of the whole solidified crust being level we have
in virtue of the assumptions of §§ 33, 34, inequalities of
6 kilometres from highest to lowest levels, or as much more
than 6 kilometres as we please to assume it.
§ 36. There must still be a small, but important, proportion
of mother liquor in the interstices between the closely packed
uncooled crystals below the wholly solidified crust. This
liquor, differing in chemical constitution from the crystals, has
its freezingpoint somewhat lower, perhaps very largely lower,
than the lowest of their meltingpoints. But, when we con
sider the mode of formation (§ 25) of the crystals from the
mother liquor, we must regard it as still always a solvent
ready to dissolve, and to redeposit, portions of the crystalline
matter, when slight variations of temperature or pressure
tend to cause such actions. Now as the specific gravity of
G2
84 Lord Kelvin on the Age of the
the liquor is less, by something like 15 per cent., than the
specific gravity of the solid crystals, it must tend to find its
way upwards, and will actually do so, however slowly, until
stopped by the already solidified impermeable crust, or until
itself becomes solid on account of loss of heat by conduction
outwards. If the upper crust were everywhere continuous
and perfectly rigid the mother liquor must, inevitably, if
sufficient time be given, find its way to the highest places of
the lower boundary of the crust, and there form gigantic
pockets of liquid lava tending to break the crust above it
and burst up through it.
§ 37. But in reality the upper crust cannot have been
infinitely strong ; and, judging alone from what we know of
properties of matter, we should expect gigantic cracks to
occur from time to time in the upper crust tending to shrink
as it cools and prevented from lateral shrinkage by the non
shrinking uncooled solid below it. When any such crack
extends downwards as far as a pocket of mother liquor
underlying the wholly solidified crust, we should have an
outburst of trap rock or of volcanic lava just such as have
been discovered by geologists in great abundance in many
parts of the world. We might even have comparatively
small portions of high plateaus of the primitive solid earth
raised still higher by outbursts of the mother liquor squeezed
out from below them in virtue of the pressure of large sur
rounding portions of the superincumbent crust. In any such
action, due to purely gravitational energy, the centre of
gravity of all the material concerned must sink, although
portions of the matter may be raised to greater heights ; but
we must leave these large questions of geological dynamics,
having been only brought to think of them at all just now by
our consideration of the earth, antecedent to life upon it.
§ 38. The temperature to which the earth's surface cooled
within a few years after the solidification reached it, must
have been, as it is now, such that the temperature at which
heat radiated into space during the night exceeds that re
ceived from the sun during the day, by the small difference
due to heat conducted outwards from within *, One year
* Suppose, for example, the cooling and thickening of the upper crust
has proceeded so far, that at the surface and therefore approximately for
a few decimetres below the surface, the rate of augmentation of tem
perature downwards is one degree per centimetre. Taking as a rough
average 005 c.G.s. as the thermal conductivity of the surface rock, we
should have for the heat conducted outwards •005 of a gramme water
thermal unit centigrade per sq. cm. per sec. (Kelvin, Math, and Phys.
Papers, vol. iii. p. 226). Hence if (ibid. p. 223) we take g^ as the
Earth as an Abode fitted for Life. 85
after the freezing of the granitic interstitial mother liquor at
the earth's surface in any locality, the average temperature
at the surface might be warmer, by 60° or 80° Cent., than if
the whole interior had the same average temperature as the
surface. To fix our ideas, let us suppose, at the end of one
year, the surface to be 80° warmer than it would be with no
underground heat : then at the end of 100 years it would be
8° warmer, and at the end of 10,000 years it would be '8 of
a degree warmer, and at the end of 25 million years it would
be *016 of a degree warmer, than if there were no under
ground heat.
§ 39. When the surface of the earth was still whitehot
liquid all round, at a temperature fallen to about 1200° Cent.,
there must have been hot gases and vapour of water above
it in all parts, and possibly vapours of some of the more
volatile of the present known terrestrial solids and liquids,
such as zinc, mercury, sulphur, phosphorus. The very rapid
cooling which followed instantly on the solidification at the
surface must have caused a rapid downpour of all the vapours
other than water, if any there were ; and a little later, rain
of water out of the air, as the temperature of the surface
cooled from red heat to such moderate temperatures as 40°
and 20° and 10° Cent., above the average due to sun heat
and radiation into the aether around the earth. What that
primitive atmosphere was, and how much rain of water fell
on the earth in the course of the first century after consoli
dation, we cannot tell for certain ; but Natural History and
Natural Philosophy give us some foundation for endeavours
to discover much towards answering the great questions,—
Whence came our present atmosphere of nitrogen, oxygen,
and carbonic acid ? Whence came our present oceans and
lakes of salt and fresh water ? How near an approximation
radiational eniissivity of rock and atmosphere of gases and watery vapour
above it radiating heat into the surrounding vacuous space (aether), we
find 8000 X 005, or 40 degrees Cent, as the excess of the mean surface
temperature above what it would be if no heat were conducted from
within outwards. The present augmentation of temperature downwards
may be taken as 1 degree Cent, per 27 metres as a rough average derived
from observations in all parts of the earth where underground temperature
has been observed. (See British Association Reports from 1868 to 1895.
The very valuable work of this Committee has been carried on for these
twentyseven years with great skill, perseverance, and success, by
Professor Everett, and he promises a continuation of his reports from
time to time.) This with the same data for conductivity and radiational
emissivity as in the preceding calculation makes 40°/2700 or 00118 u Cent.
per centimetre as the amount by which the average temperature of the
earth's surface is at present kept up by underground heat.
86 Lord Kelvin on the Age of the
to present conditions was realized in the first hundred cen
turies after consolidation of the surface ?
§ 40. We may consider it as quite certain that nitrogen
gas, carbonic acid gas, and steam, escaped abundantly in
bubbles from the mother liquor of granite, before the primi
tive consolidation of the surface, and from the mother liquor
squeezed up from below in subsequent eruptions of basaltic
rock ; because all, or nearly all, specimens of granite arid
basaltic rock, which have been tested by chemists in respect
to this question *, have been found to contain, condensed in
minute cavities within them, large quantities of nitrogen,
carbonic acid, and water. It seems that in no specimen of
granite or basalt tested has chemically free oxygen been dis
covered, while in many, chemically free hydrogen has been
found ; and either native iron or magnetic oxide of iron in
those which do not contain hydrogen. From this it might
seem probable that there was no free oxygen in the primitive
atmosphere, and that if there was free hydrogen, it was due
to the decomposition of steam by iron or magnetic oxide of
iron. Going back to still earlier conditions we might judge
that, probably, among the dissolved gases of the hot nebula
which became the earth, the oxygen all fell into combination
with hydrogen and other metallic vapours in the cooling of
the nebula, and that although it is known to be the most
abundant material of all the chemical elements constituting
the earth, none of it was left out of combination with other
elements to give free oxygen in our primitive atmosphere.
§ 41. It is, however, possible, although it might seem not
probable, that there was free oxygen in the primitive atmo
sphere. With or without free oxygen, however, but with
sunlight, we may regard the earth as fitted for vegetable life
as now known in some species, wherever water moistened the
newly solidified rocky crust cooled down below the tempera
ture of 80° or 70° of our present Centigrade thermometric
scale, a year or two after solidification of the primitive lava
had come up to the surface. The thick tough velvety coating
of living vegetable matter, covering the rocky slopes under
hot water flowing direct out of the earth at Banff (Canada) ,
lives without help from any ingredients of the atmosphere
above it, and takes from the water and from carbonic acid or
carbonates, dissolved in it, the hydrogen and carbon needed
for its own growth by the dynamical power of sunlight ; thus
* See, for example, Tilden, Proc. R. S. February 4th, 1897. "On the
Gases enclosed in Crystalline Rocks and Minerals."
f Rocky Mountains Park of Canada, on the Canadian Pacific Railway.
Earth as an Abode fitted for Life. 87
leaving free oxygen in the water to pass ultimately into the
air. Similar vegetation is found abundantly on the terraces
of the Mammoth hot springs and on the beds of the hot water
streams flowing from the Geysers in the Yellowstone National
Park of the United States. This vegetation, consisting of
confervas, all grows under flowing water at various tempera
tures, some said to be as high as 74° Cent. We cannot doubt
but that some such confervas, if sown or planted in a rivulet
or pool of warm water in the early years of the first century
of the solid earth's history, and if favoured with sunlight,
would have lived, and grown, and multiplied, and would have
made a beginning of oxygen in the air, if there had been
none of it before their contributions. Before the end of the
century, if sunheat, and sunlight, and rainfall, were suitable,
the whole earth not under water must have been fitted for
all kinds of land plants which do not require much or any
oxygen in the air, and which can find, or make, place and
soil for their roots on the rocks on which they grow ; and
the lakes or oceans formed by that time must have been
quite fitted for the life of many or ail of the species of water
plants living on the earth at the present time. The moderate
warming, both of land and water, by underground heat,
towards the end of the century, would probably be favourable
rather than adverse to vegetation, and there can be no doubt
but that if abundance of seeds of all species of the present
day had been scattered over the earth at that time, an im
portant proportion of them would have lived and multiplied
by natural selection of the places where they could best
thrive.
§ 42. But if there was no free oxygen in the primitive
atmosphere or primitive water, several thousands, possibly
hundreds of thousands, of years must pass before oxygen
enough for supporting animal life, as we now know it, was
produced. Even if the average activity of vegetable growth
on land and in water over the whole earth was, in those early
times, as great in respect to evolution of oxygen as that of
a Hessian forest, as estimated by Liebig* 50 years ago, or
of a cultivated English hayfield of the present day, a very
improbable supposition, and if there were no decay (erema
causis, or gradual recombination with oxygen) of the plants
or of portions such as leaves falling from plants, the rate of
evolution of oxygen, reckoned as three times the weight of
the wood or the dry hay produced, would be only about
* Liebig', ' Chemistry in its application to Agriculture and Physio
logy,' English, 2nd ed., edited by Playfair, 1842.
88 Lord Kelvin on the Age of the
6 tons per English acre per annum or 1^ tons per square
metre per thousand years. At this rate it would take only
1533 years, and therefore in reality a much longer time
would almost certainly be required, to produce the 2*3 tons
of oxygen which we have at present resting on every square
metre of the earth's surface, land and sea*. But probably
quite a moderate number of hundred thousand years may
have sufficed. It is interesting at all events to remark that,
at any time, the total amount of combustible material on the
earth, in the form of living plants or their remains left dead,
must have been just so much that to burn it all would take
either the whole oxygen of the atmosphere, or the excess of
oxygen in the atmosphere at the time, above that, if any,
which there was in the beginning. This we can safely say,
because we almost certainly neglect nothing considerable in
comparison with what we assert when we say that the free
oxygen of the earth's atmosphere is augmented only by
vegetation liberating it from carbonic acid and water, in
virtue of the power of sunlight, and is diminished only by
virtual burning f of the vegetable matter thus produced.
But it seems improbable that the average of the whole
earth — dry land and seabottom — contains at present coal,
or wood, or oil, or fuel of any kind originating in vegeta
tation, to so great an amount as "767 of a ton per square
metre of surface ; which is the amount at the rate of one ton
of fuel to three tons of oxygen, that would be required to
produce the 2'3 tons of oxygen per square metre of surface,
which our present atmosphere contains. Hence it seems
probable that the earth's primitive atmosphere must have
contained free oxygen.
§ 43. Whatever may have been the true history of our
atmosphere it seems certain that if sunlight was ready, the
earth was ready, both for vegetable and animal life, if not
within a century, at all events within a few hundred cen
turies after the rocky consolidation of its surface. But was
the sun ready ? The well founded dynamical theory of the
sun's heat carefully worked out and discussed by Helmholtz,
* In our present atmosphere, in average conditions of barometer and
thermometer we have, resting on each square metre of the earth's surface,
ten tons total weight, of which 77 is nitrogen and 2*3 is oxygen.
f This " virtual burning " includes ereuiacausis of decay of vegetable
matter, if there is any eremacausis of decay without the intervention of
microbes or other animals. It also includes the combination of a portion
of the food with inhaled oxygen in the regular animal economy of pro
vision for heat and power.
Earth as an Abode fitted for Life. 89
Newcomb, and myself * says NO if the consolidation of the
earth took place as long ago as 50 million years ; the solid
earth mast in that case have waited 20 or 50 million years
for the sun to be anything nearly as warm as he is at present.
If the consolidation of the earth was finished 20 or 25 million
years ago, the sun was probably ready, — though probably
not then quite so warm as at present, yet warm enough
to support some kind of vegetable and animal life on the
earth.
§ 44. My task has been rigorously confined to what,
humanly speaking, we may call the fortuitous concourse of
atoms, in the preparation of the earth as an abode fitted for
life, except in so far as 1 have referred to vegetation, as
possibly having been concerned in the preparation of an
atmosphere suitable for animal life as we now have it.
Mathematics and dynamics fail us when Ave contemplate
the earth, fitted for life but lifeless, and try to imagine the
commencement of life upon it. This certainly did not take
place by any action of chemistry, or electricity, or crystalline
grouping of molecules under the influence of force, or by any
possible kind of fortuitous concourse of atoms. We must
pause, face to face with the mystery and miracle of the
creation of living creatures.
Addendum. — May 1898.
Since this lecture was delivered I have received from
Professor RobertsAusten the following results of experiments
on the meltingpoints of rocks which he has kindly made at
my request : —
Meltingpoint. Error.
Felspar. . . 1520° C. ±30°
Hornblende . about 1400°
Mica . . . 1440° ±30°
Quartz . . . 1775° +15°
Basalt . . . about 880°
These results are in conformity with what I have said in
§§ 2628 on the probable origin of granite and basalt, as
they show that basalt melts at a much lower temperature
than felspar, hornblende, mica, or quartz, the crystalline in
gredients of granite. In the electrolytic process for pro
ducing aluminium, now practised by the British Aluminium
* See ' Popular Lectures and Addresses,' vol. i. pp. 376429, par
ticularly page 397.
90 Mr. D. L. Chapman on the
Company at their Foyers works, alumina, of which the
meltingpoint is certainly above 1700° C. or 1800° C, is
dissolved in a bath of melted cryolite at a temperature of
about 800° C. So we may imagine melted basalt to be a
solvent for felspar, hornblende, mica, and quartz at tempera
tures much below their own separate meltingpoints ; and we
can understand how the basaltic rocks of the earth may have
resulted from the solidification of the mother liquor from
which the crystalline ingredients of granite have been
deposited.
VI. On the Hate of Explosion in Gases.
By D. L. Chapman, B.A. (Oxon.)*.
THE object of the investigation of which an account is
given in this paper is the discovery of formulae to
express the maximum rates of explosion in gases and the
maximum pressure in the explosive wave.
The data which 1 propose to use are taken almost entirely
from the Bakerian Lecture of 1893, on "The Rates of
Explosion in Gases," by Prof. Dixon. The maximum
velocities of explosion given below are in all cases those
measured by Prof. Dixon or under his direction. Experi
mental conclusions only will be quoted ; for a complete
account of the experiments themselves, the reader is referred
to the abovementioned paper, and to several papers which
were subsequently published in the ' Journal of the Man
chester Literary and Philosophical Society ' and in the
; Journal of the Chemical Society. 5
Ignoring for the present all minor details connected with
particular cases, which may be more conveniently discussed
at a later stage, it is sufficient for our purpose to state at the
outset that it has been established that the maximum velocity
of explosion, in a mixture of definite composition and at fixed
temperature and pressure, has a definite value, independent
of the diameter of the tube when that diameter exceeds a
certain limit. The relations existing between temperature
and pressure and the velocity of explosion are such that an
increase of temperature causes a fall in the velocity, whereas
an increase of pressure has the reverse effect up to a certain
limit, beyond which the velocity remains constant.
For the suggestion that an explosion is in its character
essentially similar to a soundwave, we are also indebted to
Prof. Dixon ; and there is little doubt that all subsequent
advance must be made with, this suggestion as the leading
* Communicated by Prof. Dixon, F.R.S.
Rate of Explosion in Gases. 91
idea *. Although Prof. Dixon's soundwave formula has
yielded such excellent results, he has pointed out the necessity
of further a priori work in the subject.
The Rate of Explosion for an Infinite Plane Wave.
In the following attempt to establish a formula for the
velocity of explosion, I have made certain assumptions which
have not as yet received sufficient experimental confirmation ;
hut they are, I think, justified by the results. For instance,
it is assumed that, once the maximum velocity is reached, the
front of the explosion wave is of such a character that we may
suppose steady motion. This, as Prof. Schuster has pointed
out in a note to the Bakerian Lecture, is not an impossibility
when chemical change is taking place, since the implied
relation between pressure and density is possible under such
circumstances. This point, however, requires further investi
gation. The wave is assumed to be an infinite plane wave.
This assumption is justified by the fact that the diameter of
the tube is without influence on the found velocity. I propose
to limit the term u explosive wave " to the space within which
chemical change is taking place. This space is bounded by
two infinite planes. On either side of the wave are the
exploded and unexploded gases, which are assumed to have
uniform densities and velocities. The statement that the
exploded gas possesses uniform density and velocity for some
distance behind the wave requires further justification, which
can only be imperfectly given after a discussion of the general
problem.
How the true explosive wave is actually generated in
practice is a question without the scope of the present investi
gation. In order to avoid the discussion of this point, I
shall substitute for it a physical conception, which, although
unrealizable in practice, will render aid in illustrating the
views here advanced.
Let us suppose that the gas is enclosed in an infinite
cylinder ABCD, provided with a piston E, and that the
explosive wave XYZS has just started. The initial velocity
of this wave will be small ; the initial pressure along the
plane XS will also be small compared with that ultimately
attained. As the wave proceeds in the direction AB, the
piston E is supposed to follow it in such a manner that
* In the earlier researches Berthelot's theory was accepted as a working
hypothesis. It was only after the difficulties attending the measurement
of the rates of explosion in mixtures containing inert gases had been over
come that the inadequacy of Berthelot's theory became evident and the
superiority of the soundwave theory could be demonstrated.
92 Mr. D. L. Chapman on the
the pressure at EF is always kept equal to the pressure
at XS. During this process the velocity of the wave will
gradually increase, until ultimately its velocity will be
uniform, its type constant, and the exploded gas within
the area EXSF homogeneous. It is this ultimate steady
Fig. 1.
state alone which I propose to consider. During the process
just described the velocity will of course constantly increase
until it attains a maximum. After the velocity has become
uniform, and the wave permanent in type, it is obvious that
another permanent state may be reached in the following
way : — Suppose a piston is introduced immediately behind
the permanent wave, and that this piston is made to move
forward more rapidly than the previous one, the pressure and
density behind the wave will thus be increased, and after a
certain period of time another steady state will be reached.
All this is equivalent to the statement that the permanent
velocity of explosion is a function of the density of the
exploded gas.
I shall now proceed to prove the latter statement.
Since the discussion is limited to the wave of permanent
type, we may write down the condition of steady motion,
V
(1)
where V and u are the velocities of the unexploded and
exploded gas respectively, referred to coordinates moving
with a velocity — V, and v and v are the volumes of a gram
equivalent of the unexploded and exploded gas.
Take as an example cyanogen and oxygen, the explosion
of which is represented by the equation
C 2 N 2 + 2 2CO + N 2 .
224 litres + 22'4 litres = 44*8 litres + 22*4 litres.
52 grms. + 32 grins. = 56 grms. + 28 grms.
Here v = 44'8 litres, and v is the volume of carbon monoxide
Hate of Explosion in Gases. 93
and nitrogen obtained from this immediately after the
explosion.
fi = gramequivalent (in this case, 84 grms.).
From (1) and the equations of motion, we obtain
aV 2
i ? i>o= f rr(«ov) (2)*
v o
This formula of Riemann assumes a relation to subsist
between Y, p, and v at all points of the wave ; and from it
the work performed by the wave during explosion may be
calculated. • 
Work performed by the gas
fiY 2
2v<
( v v y+p {vv )
For the purpose of testing this result, it maybe shown that
the external work performed by the piston (fig. 1) is equal to
the work performed on the gas together with the gain of
kinetic energy.
The work performed on the gas
= fp  v o) 2 +Po( v o  «0 •
The gain of kinetic energy
_ (V«)V
it V 2 Yv
The external work performed by the piston
= p(v v)
_ fj,Y 2
m {vov) 2 +p (v v).
.*. External work performed by the piston
= gain of kinetic energy f work performed on the gas.
Assume that in the explosion n molecules become m mole
cules. For example, in the explosion of equal volumes of
* Rayleigh's ' Sound/ vol. ii, j Schuster's note in the Bakerian Lecture
on Explosions,
94 Mr. D. L. Chapman on the
cyanogen and oxygen, n is 2 and m is 3 :
C 2 N 2 + 2 = 2CO + N 2 .
(2 molecules) (3 molecules)
I shall now calculate the energy lost when a gas is allowed
to burn and the products of combustion are collected at the
normal temperature and pressure.
Assume that one of the gases is enclosed in the cylinder A
and the other in the cylinder B (fig. 2). These gases are forced
out, burned at C, cooled at D, and collected in the cylinder E.
The gain of energy is the work performed by the pistons
a and b ; and the loss of energy is the heat evolved at D,
together with the work performed on the piston e. The total
energy lost is the difference of these. The volume of gas in
A and B is v ; therefore the work performed by the pistons
a and b is p v Q . The volume of burnt gas is —  ; and
therefore the work performed on the piston e is ■ ^° ° .
The heat evolved at D is the heat of combustion at constant
pressure ; call it h. Let the total energy lost =H.
Then
H = A+/w>o(~l)
During an explosion the whole of this energy is retained
by the gas, and in addition to this it gains an amount of
energy equal to the work performed on the gas.
The energy of the exploded gas is therefore given by the
expression
+ energy of exploded gas at N.T.P.
n~Sf VYl
= '1 + 152(0— VqYPqV+PqVo + energy at N,T.P.
Rate of Explosion in Gases. 95
If t is the normal temperature, and t the temperature of
the gas after explosion,
mO v (t— 1 ) + energy at N.T. P. = energy of exploded gas ;
LLX 771
/. mC v (tto) = h + ^tvVoy—poV+poVo;
. iS ^ n  +t , (8)
But p = mRf,
Also, from equation (2), we get
pv = !— T (v — v)v+p v ;
v o
•' • 7T "! 7 > + ^« (*> " V o) 2  W + PoW^ i + mR«o
= 2( v ov)v+p v. . (4)
This establishes a relation between V and v. The velocity
of a permanent explosion is therefore a function of the density
of the exploded gas.
When an explosion starts its character and velocity are
continually changing until it becomes a wave permanent in
type and of uniform velocity. I think it is reasonable to
assume that this wave — i. e. the wave of which the velocity
has been measured by Prof. Dixon — is that steady wave
which possesses minimum velocity ; for, once it has become a
permanent wave with uniform velocity, no reason can be
discovered for its changing to another permanent wave
having a greater uniform velocity and a greater maximum
pressure.
This particular velocity may be discovered by eliminating v
from the equations
Y=f(v)
and dV
dv
It may be well to point out that under these circumstances
the entropy of the exploded gas is a maximum. This may be
easily shown thus : —
96 Mr. J), L, Chapman on the
The condition for maximum entropy is
= dcj> = mC v f+£dv;
w
mC v dt = —pdV)
or dt _ p
do mC v '
By differentiating (3),
but from (a) and Riemann's equation
therefore the condition of minimum velocity is equivalent to
the condition of maximum entropy*.
The following method of arriving at the approximation
v= r ^ v was suggested by Prof. Schuster, who has
shown that the method by which I arrived at the same result
is inconclusive.
Equation (4) arranged differently runs
TJ^jvVoY gV* _., r , , b
where H does not contain v.
Or putting R = CC W ,
v )v=p v I 1 + qJ — H,
^v^t?o) rc P c a/ . i c P
H— p v
c
~ ^c!
The complete expression ~r =0 leads to a quadratic ex
pression for v. Hence there are two minima or maxima.
* In any adiabatic change the entropy cannot decrease, and therefore
it tends to become a maximum.
Rate of Explosion in Gases. 97
If
v=v , Y 2 =oo,
Hence one minimum lies between these values*.
C
If H is large compared to p v pp ? V will be a minimum or
maximum when
/ \ rOp— C„ Cp + 0„ "1
is a maximum or minimum.
Writing this F: F will be zero for
G p — G v
v = oTTc^
For v = + go it will be negative ; hence between the above
values of v there will be a maximum of F or a minimum of V.
Also
2C v ^ = (C p G v )v (G p +G)v(vv )(C p + C v )
=2G P v 2(G p + G v )v.
d 2 ¥
And yt * s a ^ wa y s negative ; hence F must be a maximum
when
— u+o (5)
By eliminating v from (4) and (5) we obtain the value
of V 2 . This elimination leads to the result
2R
= jQi [{{mn)G p +mG v }G p t + (C p + C )/i],
since p Q VQ=nRt .
It is assumed throughout that the exploded gas behind the
wave remains at constant temperature and pressure, and has
dV
* The other value of v obtained from the quadratic equation j~ =0 is
much larger than v , and gives to V a very small value. It has therefore
no connexion with the wave we are considering.
Phil. Mag. 8. 5. Vol. 47. No 284. Jan. 1899. H
98 Mr. D. L. Chapman on the
a uniform velocity. Therefore during the explosion momentum
is generated by the moving piston. In an actual explosion in
a tube not provided with a piston the whole mass of gas cannot
move forward with this uniform velocity, for there would then
be a vacuum at the end of the tube where the explosion started,
and the pressure at that end would be zero, making it im
possible to account for the generation of momentum. There
is, however, no need to assume that the whole exploded gas
acquires a uniform velocity. In fact the velocity of the wave
would be the same if it were followed by a layer of exploded
gas of uniform density and velocity, and would be un affected
by any subsequent disturbance which must take place behind
the explosive wave.
It is therefore necessary to prove that behind the explosive
wave there is a layer of homogeneous gas. This evidently
must he if any disturbance behind the wave can only move
forward with a velocity less than that of the wave itself.
The forward velocity of any disturbance in the exploded
gas will be given by the sum of the velocity of the gas and
the velocity of sound in the gas.
The velocity of the gas
=Vu=Y(l?)
v Vc„+cJ
'V
The velocity of sound
_ /m&t Up
In the complete expression for V 2 the first term may be
here neglected. Also in the complete expression for t
(equation (3)) the last three terms are small. We may
therefore write
and
^Y 2
h +2^(vVo)
Employing these values, the velocity of the gas becomes
V— fere)
Hate of Explosion in Gases. 99
and the velocity of sound
./ijg7 p "."
.*. the velocity or sound + velocity of the gas
_ /2RA C/+C„ 2
"V ,, ■g*(c p +v v )
The velocity of explosion
v
2R/i (,(V+0.) !
0/(0, + 0.)
The latter is evidently greater than the former. Therefore
the layer of uniform gas behind the wave will gradually
become greater as the explosion proceeds.
Calculation of the Rates of Explosion.
In attempting to calculate the rates of explosion from the
formula there is some doubt as to what value should be adopted
for the specific heat at constant volume. This constant, has
only been directly found at comparatively low temperatures.
MM. Berthelot, Le Chatelier, and Mallard have made attempts
to find the specific heats of the elementary gases and of carbon
monoxide at high temperatures by measuring the pressure of
explosion. Berthelot arrives at the conclusion that the specific
heat at constant volume increases with the temperature, and at
4400° C. attains the value 9*6. M. Berthelot's experiments
do not, however, agree with those of MM. Le Chatelier and
Mallard, and two series of experiments conducted by the latter
experimenters do not agree with one another. The specific
heat at constant volume may, however, be calculated from
the. velocity of explosion with the aid of the proposed formula.
A few explosions have therefore been selected and the specific
heats and temperatures calculated from them ; specific heats
at intermediate temperatures being found by interpolation. It
was immediately perceived that the specific heats of O a , H 2 ,
N 2 , and CO might for all practical purposes be taken as
identical at all temperatures.
A few words are necessary regarding explosions in which
water is formed. If the specific heat of steam is taken as
f x specific heat of the diatomic gases, the found rates of ex
plosion fall below the calculated rates when the dilution with
inert gas is great, and vice versa when the dilution is small.
It is possible to account for this by two theories. The first
theory is that at high temperatures the water is dissociated,
whereas at low temperatures the combination of hydrogen and
oxygen is complete. The second theory is that the specific heat
H 2
100
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Rate of Explosion in Gases.
101
of steam rises more rapidly with the temperature than the spe
cific heat of the diatomic gas. The theory of dissociation is
rendered improbable by the fact that dilution of electrolytic
gas with oxygen lowers the rates a little more than dilution
with an equal volume of nitrogen. The adoption of such an
hypothesis would render it necessary for us to suppose that the
chemical reaction does not proceed to its limit. Moreover, it
would make it difficult to calculate the rates whenever steam
is formed, for it would then be impossible, with our present
knowledge, to say how far the chemical reaction would pro
ceed in any particular case. We are therefore encouraged to
test the first theory, i. e. that the specific heat of steam rises
more rapidly with the temperature than that of the diatomic
gases. The specific heat of steam at different temperatures
has therefore been calculated from a few selected rates, as in
the case of the elementary gases; and the values thus found
are used to calculate the other rates. The results are given
below. (Table II.)
Table II. — Specific Heats at Different Temperatures.
w = specific heat of water, g = specific heat of diatomic gases.
t.
5600.
5500.
5400.
5300.
5200.
5100.
5000.
4900.
4800.
w
9
7850
7839
7828
7817
7806
7795
7784
7773
7762
t.
w
9
4700.
4600.
7740
4500.
4400.
4300.
4200.
4100.
4000.
3900.
7751
7*729
7718
14750
7707
14625
7696
14467
7685
14297
7674
14125
7663
t.
3800.
3700.
3600.
3500.
3400.
3300.
3200.
3100.
3000.
w
9
13938
7652
13750
7641
13547
7630
13344
7619
13102
7608
12850
7597
12560
7586
12250
7575
11891
7564
t.
w
9
2900.
11503
7553
2800.
2700.
2600.
2500.
2400.
2300.
2200.
2100.
11040
7542
10578
7531
10172
7520
9797
7509
9484
7498
9203
7487
9000
7476
8828
7*466
102
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104 Dr. 0. Barus on the Aqueous Fusion of Glass.
On referring to the explosion of ethylene with excess of
oxygen it is seen that C0 2 is not completely dissociated until
a temperature of 3500° C. is reached. In all cases the tem
perature of explosion of cyanogen with excess of oxygen is
above this, and therefore C0 2 is never formed.
The Pressure of Explosion.
The maximum pressure of explosion may be calculated with
the aid of the two formulae
llY 2
and
These two equations lead to the formula
/*V 2 C, ^
The pressure for an explosion of equal volumes of cyanogen
and oxygen calculated from this formula is 57 atmospheres.
Jones and Bower "* by breaking glass tubes obtain the value
58 atmospheres
VII. The Aqueous Fusion of Glass, its Relation to Pressure
and Temperature. First Paper. By Carl Barus f.
SOME time ago I published J a series of results due to the
action of hot water at 185° on glass, the water being
kept liquid by pressure. It was shown that the water con
tained in sealed capillary glass tubes increased in compressi
bility while it steadily diminished in bulk, as described in the
subjoined summary of two consistent experiments with
different tubes. During the observations the column soon
became turbid, but it remained translucent enough to admit
of measurement. As the action at 185° proceeded, the
length of the thread of water decreased. This thread was
contained within the walls of the tube between two terminal
threads of mercury (the lower being movable and trans
mitting pressure), and therefore decrease in the length of the
thread can only mean contraction of volume of the system of
glass and water in contact. The results are as follows : —
6 denoting the temperature of the capillary thread (main
tained constant by a transparent vapourbath) ; t the time
* Journal of the Manchester Lit. and Phil. Soc. 1898.
t Communicated by the Author.
 Barus : American Journal of Science, xli. p. 110 (1891).
Capillary tube, with appurtenances,
for measuring the Compressibility
of Liquids.
s ■ * —
6"  —
AABB. Flange screwing to compressionpump.
a be. Capillary tube containing thread of water between visible
threads of mercury. Ends of threads at S and S'.
G G. Waterbath to cool paraffin plug of capillary tube.
eeee. Annular vapourbath of glass, containing charge, hh, of aniline
oil, kept in ebullition by the ringburner HE. Non
conducting jacketing of tube and screens not shown.
D tubulure for condenser, T for thermometer.
Observations made with the cathetometer observing each mercury
meniscus along the lines of sight S, S' , through the clear walls of the glass
vapourbath eeee.
106
Dr. C. Barus on the
of the observation counting from the beginning of ebul
lition in the vapourbath ; v the total increment of volume
due to the thermal expansion, V the total volume, so that
v/Y is the mean expansion per uuit of volume at 185° ; /3 the
mean compressibility within 300 atmospheres.
Table showing Thermal Expansion and Compressibility of
Silicated Water at 185° and 20 to 300 atmospheres.
Diameter of tube '045 centim. Length of column of
water at 24°, 14 centim.
e.
v/V . 10 3 .
(3 . 10 G .
t.
e.
v/V . 10 3 .
/3.10 6 .
t.
°c.
min.
°c.
mm.
24
±0
44
185
444
141
40
185
+ 103
77
18
185
+ 27
163
45
185
86
97
25
185
+05
184
50
185
75
112
30
185
15
221
55
185
60
125
35
185
29
60
At the conclusion of the experiment the thread was solid,
as I supposed, at high pressure (300 atm.), though not
apparently so at low pressure. This was inferred since the
mercury thread advancing under pressure did not, on removal
of pressure, return again as a whole, but broke into small
parts in a way to make further measurement without
immediate value*. On breaking the tube apart after cooling,
and examining it under the microscope, the capillary canal
was found to be nearly, if not quite, filled with a white glassy
incrustation. This shows that the glass swells in marked
degree on hydration, whereas the combined volume of glass
and water put into action, simultaneously contracts. If the
values of v/Y given in the table be examined, it appears that
whereas the original volume increment per unit of volume is
greater than *103 for the rise of temperature from 24° to 185°
at 20 atmospheres, this increment has nearly vanished after
50 minutes of reaction. The thread at 185° is now only as
long as it was at 24°. After 60 minutes of reaction it is even
markedly shorter at 185° than at 24°, pressure remaining
constant throughout.
At the same time the compressibility, /3, of the silicated
* A succeeding paper will take up the research from this point
onward.
Aqueous Fusion of Glass. 107
water at 185° is found to increase regularly from '000077
near the beginning of the experiment, to over '000221, or
more than three times its initial value on the same isothermal
(185°). This result is wholly unexpected, since without
exception the effect of solution is a decrease of the compressi
bility of the solvent, in proportion as more body is dissolved.
Silicated water in the present experiment shows the reverse
effect. Now although the hydration increases the volume of
the glass, the gradual choking of the capillary canal goes on
uniformly from top to bottom of the thread of water. Hence,
since the bore is diminished at the same rate throughout the
wetted tube, the observations for compressibility would remain
to the same degree unchanged, ccet. par. Supposing that fine
particles of the glass were broken off* and gradually accumu
lated on the mercury meniscus near the bottom of the thread
of water, it would be possible to account for the data for /3
cited, in consideration of the gradual constriction of the
thread near the bottom. In such a case, however, the com
pressing thread of mercury would not have advanced and
retreated through this debris with the observed regularity.
In general mere stoppage and clogging would have been
noticed in duplicate experiments f. I also made correlative
experiments with saturated solutions of zinc sulphate in water
and naphthalene in alcohol. In both cases markedly increased
compressibilities were possible in a turbid column, and due to
the precipitation of part of the dissolved salt, isothermally,
by pressure. During compression a part of the dissolved body
is changed from the liquid to the solid state by pressure, and
hence the apparent increase of compressibility.
From this point of view I have endeavoured to account
preliminarily for the observed regular increase of j3 given in
the table, though I confess some reluctance to this explana
tion : I have supposed that the dissolved silicate is precipitated
out of solution by pressure and redissolved on removing
pressure, thus producing accentuated compressibility ; that
this effect increases as more silicate is taken up in solution,
until finally the whole thread becomes too viscous for further
observation. However this may be, the fact of a regularly
and enormously increased compressibility remains as colla
teral evidence of the stage of progress of the reaction.
2. There is a final result to be obtained from this experi
ment, and it is to this that my remarks chiefly apply. The
reaction of the water on the glass must be along the surface
* This does not occur. See below.
t Thus for instance the thickness of the thread of mercury seen in the
cathetometer did not seem to diminish.
108 Dr. 0. Barus on the Aqueous Fusion of Glass.
of contact of both bodies. For a given length f thread this
surface decreases as the radius, r, of the tube. On the other
hand, the volume of water decreases as the square of the radius,
r, of the capillary tube. In fact, if V be the volume and S
the surface for a given length of thread, S/V = 2/r. Let a be
the rate of absorption of water in glass, i. e., the number of
cub. centim. of water absorbed per square centim. of surface
of contact, per minute. If v is the volume absorbed by S
cm. 2 per minute, v = a S and therefore v/V = 2 ajr. Hence, if
r is large, the apparent effect of absorption vanishes ; but in
proportion as r is smaller, or as the tube becomes more finely
capillary, the effect of absorption will become more obvious
to the eye. In other words, the length of the column of water
included between the two terminal threads of mercury will
decrease faster for small values of the capillary radius.
In the above results v/V taken directly from the table is
about *003 cub. centim. per minute. The diameter of the tube
measured microscopically was found to be about '045 centim.
Therefore a = '000034 cub. centim. is the volume of water
absorbed per square centim. of surface of contact, per minute,
at 185°. This is about 180 kg., per sq. metre, per year, at 185°.
True the phenomenon is not quite so simple as here com
puted, for as the action proceeds the water holds more body
in solution, the area of unchanged glass increases, and
possibly the liquid must diffuse or percolate through the
layer of opalescent accretion to reach it. As against the
seriousness of this consideration, one may allude to the
regularity of the above results in the lapse of time and the
occurrence of a reaction rather accelerated as time increases.
In view of the large surfaces of reaction available even in
small bulks of porous or triturated rock and the fact that the
intensity of the reaction increases rapidly with temperature,
I cannot but regard this result as important. Direct experi
ments * have been made with care to detect a possible thermal
effect (rise of temperature) of the action of water on hot glass,
but thus far without positive results. The difficulties of such
experiments are very great. To insure chemical reaction,
they must be made with superheated water under pressure,
with allowances for heat conduction &c, all of wdiich make the
measurement of small increments of temperature very
uncertain. If, however, rise of temperature may be associated
with the marked contraction of volume in the system water
glass specified, one may note, in the first place, that for a
* G. F. Becker : Monographs U. S. Geolog. Survey, No. III., 1882.
I have since made similar experiments with superheated water (200°).
Uranium Radiation and Electrical Conduction produced. 109
capillary canal about one half millimetre in diameter, the
absorption of water is as great as 18 per cent, of the volume
contained per hour. In finely porous rock correspondingly
larger absorptions are to be anticipated. Again, the tempera
tures and pressures given in the above experiments would be
more than reached by a column of water penetrating a few
miles below the earth's surface. Finally, the action of water
on silicates will be accelerated in proportion as higher
temperatures are entered with increasing terrestrial depth.
Eventually, therefore, heat must be evolved more rapidly than
it is conducted away.
With the above proviso, one may reasonably conclude that
the action of hot water on rock within the earth constitutes a
furnace whose efficiency increases in marked degree with the
depth of the seat of reaction below sealevel.
Brown University,
Providence, U.S.A.
VIII. Uranium Radiation and the Electrical Conduction pro
duced by it. By E. Butherford, M.A., B.Sc, formerly
1851 Science Scholar, Coutts Trotter Student, Trinity
College, Cambridge ; McDonald Professor of Physics,
McGill University, Montreal*.
THE remarkable radiation emitted by uranium and its
compounds has been studied by its discoverer, Becquerel,
and the results of his investigations on the nature and pro
perties of the radiation have been given in a series of papers
in the Comptes Rendus^. He showed that the radiation, con
tinuously emitted from uranium compounds, has the power
of passing through considerable thicknesses of metals and
other opaque substances ; it has the power of acting on a
photographic plate and of discharging positive and negative
electrification to an equal degree. The gas through which
the radiation passes is made a temporary conductor of electri
city and preserves its power of discharging electrification for
a short time after the source of radiation has been removed.
The results of Becquerel showed that Bontgen and uranium
radiations were very similar in their power of penetrating
solid bodies and producing conduction in a gas exposed to
them ; but there was an essential difference between the two
types of radiation. He found that uranium radiation could
be refracted and polarized, while no definite results showing
&
* Communicated by Prof. J. J. Thomson, F.K.S.
t C. R. 1896, pp. 420, 501, 559, 689, 762, 1086 ; 1897, pp. 43S, 800.
110 Prof. E. Rutherford on Uranium Radiation and
polarization or refraction have been obtained for Rontgen
radiation.
It is the object of the present paper to investigate in more
detail the nature of uranium radiation and the electrical
conduction produced. As most of the results obtained have
been interpreted on the ionizationtheory of gases which was
introduced to explain the electrical conduction produced by
Rontgen radiation, a brief account is given of the theory and
the results to which it leads.
In the course of the investigation, the following subjects
have been considered: —
§ 1. Comparison of methods of investigation.
§ 2. Refraction and polarization of uranium radiation.
§ 3. Theory of ionization of gases.
§ 4. Complexity of uranium radiation.
§ 5. Comparison of the radiation from uranium and its
compounds.
§ 6. Opacity of substances for the radiation.
§ 7. Thorium radiation.
§ 8. Absorption of radiation by gases.
§ 9. Variation of absorption with pressure.
§ 10. Effect of pressure of the gas on the rate of discharge.
§ 11. The conductivity produced in gases by complete
absorption of the radiation.
§ 12. Variation of the rate of discharge with distance
between the plates.
§ 13. Rate of recombination of the ions.
§ 14. Velocity of the ions.
§ 15. Fall of potential between two plates.
§ 16. Relation between the current through the gas and
electromotive force applied.
§ 17. Production of charged gases by separation of the ions.
§ 18. Discharging power of fine gauzes.
§ 19. General remarks.
§ 1. Comparison of Methods of Investigation.
The properties of uranium radiation may be investigated
by two methods, one depending on the action on a photo
graphic plate and the other on the discharge of electrification.
The photographic method is very slow and tedious, and admits
of only the roughest measurements. Two or three days'
exposure to the radiation is generally required to produce any
marked effect on the photographic plate. In addition, when
we are dealing with very slight photographic action, the
the Electrical Conduction produced by it. Ill
fogging of the plate, during the long exposures required, by
the vapours of substances * is liable to obscure the results.
On the other hand the method of testing the electrical dis
charge caused by the radiation is much more rapid than the
photographic method, and also admits of fairly accurate
quantitative determinations.
The question of polarization and refraction of the radiation
can, however, only be tested by the photographic method.
The electrical experiment (explained in § 2) to test refraction
is not very satisfactory.
§ 2. Polarization and Refraction.
The almost identical effects produced in gases by uranium
and Rontgen radiation (which will be described later) led me
to consider the question whether the two types of radiation
did not behave the same in other respects.
In order to test this, experiments were tried to see if
uranium radiation could be polarized or refracted. Becquerel f
had found evidence of polarization and refraction, but in
repeating experiments similar to those tried by him, I have
been unable to find any evidence of either. A large number
of photographs by the radiation have been taken under
various conditions, but in no case have I been able to observe
any effect on the photographic plate which showed the presence
of polarization or refraction.
In order to avoid fogging of the plate during the long
exposures required, by the vapours of substances, lead was
employed as far as possible in the neighbourhood of the
plate, as its effect on the film is very slight.
A brief account will now be given of the experiments on
refraction and polarization.
Refraction. — A thick lead plate was taken and a long
narrow slit cut through it ; this was placed over a uniform
layer of uranium oxide ; the arrangement was then equivalent
to a line source of radiation and a slit. Thin prisms of glass,
aluminium, and paraffinwax were fixed at intervals on the
lead plate with their edges just covering the slit. A photo
graphic plate was supported 5 mms. from the slit. The plate
was left for a week in a dark box. On developing a dark
line was observed on the plate. This line was not appreciably
broadened or displaced above the prisms. Different sizes of
slits gave equally negative results. If there was any appreci
able refraction we should expect the image of the slit to be
displaced from the line of the slit.
* Russell, Proc. Roy. Soc. 1897. f C, R. 1896, p. 559.
112 Prof. E. Rutherford on Uranium Radiation and
Becquerel* examined the opacity of glass for uranium
radiation in the solid and also in a finelypowdered state by
the method of electric leakage, and found that, if anything,
the transparency of the glass for the radiation was greater in
the finelydivided than in the solid state. 1 have repeated
this experiment and obtained the same result. As Becquerel
stated, it is difficult to reconcile this result with the presence
of refraction.
Polarization. — An arrangement very similar to that used
by Becquerel was employed. A deep hole was cut in a thick
lead plate and partly filled with uranium oxide. A small
tourmaline covered the opening. Another small tourmaline
was cut in two and placed on top of the first, so that in
one half of the opening the tourmalines were crossed and in
the other half uncrossed. The tourmalines were very good
optically. The photographic plate was supported 1 to 3 mm.
above the tourmalines. The plate was exposed four days, and
on developing a black circle showed up on the plate, but in
not one of the photographs could the slightest difference in
the intensity be observed. Becquerel f stated that in his
experiment the two halves were unequally darkened, and
concluded from this result that the radiation was doubly
refracted by tourmaline, and that the two rays were unequally
absorbed.
§3. Theory of Ionization.
To explain the conductivity of a gas exposed to Rontgen
radiation, the theory % has been put forward that the rays in
passing through the gas produce positively and negatively
charged particles in the gas, and that the number produced
per second depends on the intensity of the radiation and the
pressure.
These carriers are assumed to be so small that they will
move with a uniform velocity through a gas under a constant
potential gradient. The term ion was given to them from
analogy with electrolytic conduction, but in using the term
it is not assumed that the ion is necessarily of atomic dimen
sions ; it may be a multiple or submultiple of the atom.
Suppose we have a gas between two plates exposed to the
radiation and that the plates are kept at a constant difference
of potential. A certain number of ions will be produced
per second by the radiation and the number produced will in
general depend on the pressure of the gas. Under the electric
* C. It. 1896, p. 559. t C. R. 1896, p. 559.
% J. J. Thomson and E. Rutherford, Phil. Mag, Nov. 1896.
the Electrical Conduction produced by it. 113
field the positive ions travel towards the negative plate and
the negative ions towards the other plate, and consequently
a current will pass through the gas. Some of the ions
will also recombine, the rate of recombination toeing pro
portional to the square of the number present. The current
passing through the gas for a given intensity of radiation will
depend on the difference of potential between the plates, but
when the potentialdifference is greater than a certain value
the current will reach a maximum. When this is the case
all the ions are removed by the electric field before they
can recombine.
The positive and negative ions will be partially separated
by the electric field, and an excess of ions of one sign may be
blown away, so that a charged gas will be obtained. If the ions
are not uniformly distributed between the plates, the potential
gradient will be disturbed by the movement of the ions.
If energy is absorbed in producing ions, we should expect
the absorption to be proportional to the number of ions pro
duced and thus depend on the pressure. If this theory be
applied to uranium radiation we should expect to obtain the
following results : —
(1) Charged carriers produced through the volume of the
(2) Ionization proportional to the intensity of the radiation
and the pressure.
(3) Absorption of radiation proportional to pressure.
(4) Existence of saturation current.
{>)) Rate of recombination of the ions proportional to the
square of the number present.
(6) Partial separation of positive and negative ions.
(7) Disturbance of potential gradient under certain con
ditions between two plates exposed to the radiation.
The experiments now to be described sufficiently indicate
that the theory does form a satisfactory explanation of the
electrical conductivity produced by uranium radiation.
In all experiments to follow, the results are independent of
the sign of the charged plate, unless the contrary is expressly
stated.
§ 4. Complex Nature of Uranium Radiation.
Before entering on the general phenomena of the conduction
produced by uranium radiation, an account will be given of some
experiments to decide whether the same radiation is emitted
by uranium and its compounds and whether the radiation is
Phil. Mag. S. 5. Vol. 47. No. 284. Jan. 1899. I
114 Prof. E. Rutherford on Uranium Radiation and
homogeneous. Rontgen and others have observed that the
tfrays are in general of a complex nature, including rays of
wide differences in their power of penetrating solid bodies.
The penetrating power is also dependent to a large extent on
the stage of exhaustion of the Crookes tube.
In order to test the complexity of the radiation, an electrical
method was employed. The general arrangement is shown
in fig. 1.
Ffc. 1.
The metallic uranium or compound of uranium to be
employed was pow T dered and spread uniformly over the centre
of a horizontal zinc plate A, 20 cm. square. A zinc plate B,
20 cm. square, w y as fixed parallel to A and 4 cm. from it.
Both plates were insulated. A was connected to one pole of
a battery of 50 volts, the other pole of which was to earth ;
B was connected to one pair of quadrants of an electrometer,
the other pair of which was connected to earth.
Under the influence of the uranium radiation there was a
rate of leak between the two plates A and B. The rate of
movement of the electrometerneedle, when the motion was
steady, was taken as a measure of the current through the
gas.
Successive layers of thin metal foil were then placed over
the uranium compound and the rate of leak determined for
each additional sheet. The table (p. 115) shows the results
obtained for thin Dutch metal.
In the third column the ratio of the rates of leak for each
additional thickness of metal leaf is given. Where two
thicknesses were added at once, the square root of the observed
ratio is taken, for three thicknesses the cube root. The
table shows that for the first ten thicknesses of metal the rate
of leak diminished approximately in a geometrical progression
as the thickness of the metal increased in arithmetical pro
gression.
the Electrical Conduction produced by it,
Thickness of Metal Leaf 00008 cm.
Layer of Uranium Oxide on plate.
115
Number of
Leak per nrin. in
Ratio for each
Layers.
scaledivisions.
layer.
91
1
77
•85
2
60
•78
a
49
•82
4
42
•86
5
33
•79
6
247
•75
8
154
•79
10
91
•77
13
58
•86
It will be shown later (§ 8) that the rate of leak between
two plates for a saturating voltage is proportional to the
intensity of the radiation after passing through the metal.
The voltage of 50 employed was not sufficient to saturate the
gas, but it was found that the comparative rates of leak under
similar conditions for 50 and 200 volts between the plates
were nearly the same. When we are dealing with very
small rates of leak, it is advisable to employ as small a voltage
as possible, in order that any small changes in the voltage
of the battery should not appreciably affect the result. For
this reason the voltage of 50 was used, and the comparative
rates of leak obtained are very approximately the same as
for saturating electromotive forces.
Since the rate of leak diminishes in a geometrical pro
gression with the thickness of metal, we see from the above
statement that the intensity of the radiation falls off in a
geometrical progression, i. e. according to an ordinary absorp
tion law. This shows that the part of the radiation considered
is approximately homogeneous.
With increase of the number of layers the absorption
commences to diminish. This is shown more clearly by using
uranium oxide with layers of thin aluminium leaf (see table
p. 116).
It will be observed that for the first three layers of aluminium
foil, the intensity of the radiation falls on according to the
ordinary absorption law, and that, after the fourth thickness,
the intensity of the radiation is only slightly diminished by
adding another eight layers.
12
116 Prof. E. Kutherford on Uranium Radiation and
Thickness of Aluminium foil *0005 cm.
Number of Layers
Leak per min. in.
Eatio.
of Aluminium foil.
scaledivisions.
182
1
77
•42
2
33
•43
3
146
•44
4
94
•65
12
7
The aluminium foil in this case was about '0005 cm. thick,
so that after the passage of the radiation through '002 cm. of
aluminium the intensity of the radiation is reduced to about
2V of its value. The addition of a thickness of *001 cm. of
aluminium has only a small effect in cutting down the rate
of leak. The intensity is, however, again reduced to about
half of its value after passing through an additional thickness
of '05 cm., which corresponds to 100 sheets of aluminium
foil.
These experiments show that the uranium radiation is
complex, and that there are present at least two distinct types
of radiation — one that is very readily absorbed, which will be
termed for convenience the a radiation, and the other of a
more penetrative character, which will be termed the ft
radiation.
The character of the j3 radiation seems to be independent
of the nature of the filter through which it has passed. It
was found that radiation of the same intensity and of the same
penetrative power was obtained by cutting off the a radiation
by thin sheets of aluminium, tinfoil, or paper. The /S radiation
passes through all the substances tried with far greater facility
than the a radiation. For example, a plate of thin cover
glass placed over the uranium reduced the rate of leak to ^
of its value; the j3 radiation, however, passed through it
with hardly any loss of intensity.
Some experiments with different thicknesses of aluminium
seem to show, as far as the results go, that the /3 radiation is
of an approximately homogeneous character. The following
table gives some of the results obtained for the ft radiation
from uranium oxide : —
the Electrical Conduction 'produced by it.
fi Radiation.
117
Thickness of
Aluminium.
Rate of Leak.
•005
•028
•051
•09
1
•68
•48
•25
The rate of leak is taken as unity after the a radiation has
been absorbed by passing through ten layers of aluminium
foil. The intensity of the radiation diminishes with the
thickness of meted traversed according to the ordinary
absorption Jaw. It must be remembered that when we are
dealing with the ft radiation alone, the rate of leak is in
general only a few per cent, of the leak due to the a radiation,
so that the investigation of the homogeneity of the /3 radiation
cannot be carried out with the same accuracy as for the a,
radiation. As far, however, as the experiments have gone,
the results seem to point to the conclusion that the /3 radiation
is approximately homogeneous, although it is possible that
other types of radiation of either small intensity or very
great penetrating power may be present.
§ 5. Radiation emitted by different Compounds of Uranium.
All the compounds of uranium examined gave out the two
types of radiation, and the penetrating power of the radiation
for both the a and /3 radiations is the same for all the compounds.
The table (p. 118) shows the results obtained for some of
the uranium compounds.
Fig. 2 shows graphically some of the results obtained for
the various uranium compounds. The ordinates represent
rates of leak, and the abscissas thicknesses of aluminium
through which the radiation has passed.
The different compounds of uranium gave different rates
of leak, but, for convenience of comparison, the rate of leak
due to the uncovered salt is taken as unity.
It will be seen that the rate of decrease is approximately
the same for the first layer of metal, and that the rate of
decrease becomes much slower after four thicknesses of foil.
The rate of leak due to the /3 radiation is a different
proportion of the total amount in each case. The uranium
118 Prof. E. Rutherford on Uranium Radiation and
Thickness of Aluminium foil *0005 cm.
Number of
Layers of
Aluminium foil.
Proportionate Rate of Leak.
Uranium
metal.
Uranium
Nitrate.
Uranium
Oxide.
Uranium
Potassium
Sulphate.
1
2
3
4
5
12
1
•51
•35
•15
1
•43
•28
•17
•15
•125
1
•42
•18
•08
•05
•04
1
•42
•27
•17
•12
•11
Fig. 2.
1
\
w
\
=>
5
\\
X 5
\\
V \
•s
\
\
^^
yvww
©
UffANh
;m r'orfi
ss/cw Si
1 0X/D£
UPHAT£
10 E0 30 40
Thickness of Aluminium : each division 00012 cm.
the Electrical Conduction produced by it.
119
metal was used in the form of powder, and a smaller area of
it was used than in the other cases. For the experiments on
uranium oxide a thin layer of fine powder was employed,
and we see, in that case, that the ft radiation bears a much
smaller proportion to the total than for the other compounds.
When a thick layer of the oxide was used there was, how
ever, an increase in the ratio, as the following table shows : —
Number of layers
of Aluminium foil.
Rate of Leak.
Thin layer of
Uranium Oxide.
Thick layer of
Uranium Oxide.
1
1
1
•42
5
2
•18
4
•05
•12
8
•113
12
•04
...
18
...
•11
The amount of the a radiation depends chiefly on the
surface of the uranium compound, while the ft radiation
depends also on the thickness of the layer. The increase of
the rate of leak due to the ft radiation with the thickness of
the layer indicates that the ft radiation can pass through a
considerable thickness of the uranium compound. Experi
ments showed that the leak due to the a radiation did not
increase much with the thickness of the layer. I did not,
however, have enough uranium salt to test the variation of
the rate of leak due to the ft radiation for thick layers.
The rate of leak from a given weight of uranium or
uranium compound depends largely on the amount of surface.
The greater the surface, the greater the rate of leak. A small
crystal of uranium nitrate was dissolved in water, and the
water then evaporated so as to deposit a thin layer of the
salt over the bottom of the dish. This gave quite a large
leakage. The leakage in such a case is due chiefly to the
a radiation.
Since the rate of leak due to any uranium compound
depends largely on its amount of surface, it is difficult to
compare the quantity of radiation given out by equal amounts
of different salts : for the result will depend greatly on the
120 Prof. E. Rutherford on Uranium Radiation and
state of division of the compound, It is possible that the
apparently very powerful radiation obtained from pitchblende
by Curie * may be partly due to the very fine state of division
of the substance rather than to the presence of a new and
powerful radiating substance.
The rate of leak due to the /3 radiation is, as a rule, small
compared with that produced by the a radiation. It is
difficult, however, to compare the relative intensities of the
two kinds. The a radiation is strongly absorbed by gases
(§8), while the /3 radiation is only slightly so. It will be.
shown later (§ 8) that the absorption of the radiation by the
gas is approximately proportional to the number of ions
produced. . If therefore the j3 radiation is only slightly
absorbed by the gas, the number of ions produced by it is
small, i. e. the rate of leak is small. The comparative rates
of leak due to the a and /3 radiation is thus dependent on the
relative absorption of the radiations by the gas as well as on
the relative intensity.
The photographic actions of the a and /3 radiations have
also been compared. A thin uniform layer of uranium oxide
was sprinkled over a glass plate ; one half of the plate was
covered by a piece of aluminium of sufficient thickness to prac
tically absorb the a radiation. The photographic plate was fixed
about 4 mm. from the uranium surface. The plate was exposed
48 hours, and, on developing, it was found that the darkening of
the two halves w T as not greatly different. On the one half of the
plate the action was due to the j3 radiation alone, and on the
other due to the a\ and ft radiations together. Except when
the photographic plate is close to the uranium surface, the
photographic action is due principally to the j3 radiation.
§ 6. Transparency of Substances to the two Types of
Radiation.
If the intensity of the radiation in traversing a substance
diminishes according to the ordinary absorption law, the
ratio r of the intensity of the radiation after passing through
a distance d of the substance to the intensity when the
substance is removed is given by
r = e ,
where X is the coefficient of absorption and e=2*7.
In the following table a few values of X are given for the
a and @ radiations, assuming in each case that the radiation is
* C. R. July 1898, p. 175.
the Electrical Conduction produced by it.
121
simple and that the intensity falls off according to the above
law : —
Substance.
\
for the a radiation.
X
for the /3 radiation.
Dutch metal
Aluminium
Tinfoil
2700
1600
2650
15
108
49
97
240
56
Copper
Silver
Glass
The above results show what a great difference there is in
the power of penetration of the two types of radiation. The
transparency of aluminium for the (3 radiation is over 100
times as great as for the a radiation. The opacity of the
metals aluminium, copper, silver, platinum for the /3 radiation
follows the same order as their atomic weights. Aluminium
is the most transparent of the metals used, but glass is more
transparent than aluminium for the (3 radiation. Platinum
has an opacity 16 times as great as aluminium. For the
a radiation, aluminium is more transparent than Dutch metal
or tinfoil.
For a thickness of aluminium "09 cm. the intensity of the
ft radiation was reduced to '25 of its value ; for a thickness
of copper '03 cm. the intensity was reduced to *23 of its
value. These results are not in agreement with some given
by Becquerel * 9 who found copper was more transparent than
aluminium for uranium radiation.
The /3 radiation has a penetrating power of about the same
order as the radiation given out by an average #ray bulb.
Its power of penetration is, however, much less than for the
rays from a " hard " bulb. The a radiation, on the other
hand, is far more easily absorbed than rays from an ordinary
bulb, but is very similar in its penetrating power to the
secondary radiation  sent out when .zrays fall upon a metal
surface.
It is possible that the a radiation is a secondary radiation
set up at the surface of the uranium by the passage of the
/3 radiation through the uranium, in exactly the same way
* C. R. 1896, p. 763.
f Perrin, C. R. cxxiv. p. 455 ; Sagnac, C. R. 1898.
122 Prof. E. Rutherford on Uranium Radiation and
as a diffuse radiation is produced at the surface of a metal by
the passage of Rontgenrays through it. There is not, how
ever, sufficient evidence at present to decide the question.
§ 7. Thorium Radiation.
While the experiments on the complex nature of uranium
radiation were in progress, the discovery * that thorium
and its salts also emitted a radiation, which had general
properties similar to uranium radiation, was announced. A
few experiments were made to compare the types of radiation
emitted by uranium and thorium.
The nitrate and the sulphate of thorium were used and gave
similar results, although the nitrate appeared to be the more
active of the two. The leakage effects due to these salts were
of quite the same order as those obtained for the uranium
compounds; but no satisfactory quantitative comparison can
be made between the uranium and thorium salts as the amount
of leak depends on the amount of surface and thickness of
the layer.
It was found that thorium nitrate when first exposed to the
air on a platinum plate was not a steady source of radiation,
and for a time the rate of leak varied very capriciously, being
sometimes five times as great as at others. The salt was very
deliquescent, but after exposure of some hours to the atmo
sphere the rate of leak became more constant and allowed of
rough comparative measurements. Thorium sulphate was
more constant than the nitrate.
The absorption of the thorium radiation was tested in the
same way as for uranium radiation. The following table
gives some of the results. The aluminium foil was of the
same thickness ('0005 cm.) as that used in the uranium
experiments : — ■
Number of Layers
of Aluminium foil.
Leak per minute in
scaledivisions.
200
4
94
8
37
12
19
17
75
* G. C. Schmidt, Wied. Annal. May 1898.
the Electrical Conduction produced by it. 123
The curve showing the relation between the rate of leak
and the thickness of the metal traversed is shown in fig. 2
(p. 118), together with the results for uranium.
It will be seen that thorium radiation is different in pene
trative power from the a radiation of uranium. The radiation
will pass through between three and four thicknesses of alu
minium foil before the intensity is reduced to onehalf, while
with uranium radiation the intensity is reduced to less than a
half after passing through one thickness of foil.
With a thick layer of thorium nitrate it was found that the
radiation was not homogeneous, but rays of a more penetrative
kind were present. On account of the inconstancy of thorium
nitrate as a source of radiation, no accurate experiments have
been made on this point.
The radiations from thorium and uranium are thus both
complex, and as regards the a type of radiation are different
in penetrating power from each other.
In all the experiments on uranium and thorium, care was
taken that no stray radiation was present which would obscure
the results. Such precautions are very necessary when the
rate of leak, due to the radiation transmitted through a con
siderable thickness of metal, is only a small percentage of the
total. The method generally employed was to cover the layer
of active salt with the metal screen, and then place in position
over it a large sheet of lead with a rectangular hole cut in it
of smaller area than that of the layer of salt. The lead was
pressed tightly down, and the only radiation between the
parallel plates had to pass through the metal screen, as the
lead was too thick to allow any to go through.
§8. Absorption of Uranium Radiation by Gases.
The a. radiation from uranium and its compounds is rapidly
absorbed in its passage through gases. The absorption for
hydrogen, air, and carbonic acid was determined, and was
found to be least in hydrogen and greatest in carbonic acid.
To show the presence of absorption, the following arrange
ment (fig. 3) was used : —
A layer of uraniumpotassium sulphate or uranium oxide
was spread uniformly over a metal plate P, forming a lamella
of 11 cm. diameter. A glass vessel Gr, 12 cm. in diameter, was
placed over the layer. Two parallel metal plates A and B,
1*5 cm. apart, were insulated from each other by ebonite
rods. A circular opening 7 cm. in diameter was cut in the
plate A, and the opening covered by a sheet of aluminium
foil O0005 cm. thick. The plate B was connected through a
124 Prof. E. Rutherford on Uranium Radiation and
rod R to a screw adjustment, S, so that the condenser AB
could be moved as a whole parallel to the baseplate. The
system AB was adjusted parallel to the uranium surface and
did not rotate with the screw. The rod R passed through
<*£"*> x), h
h
£/)RTH
a short glass tube fixed in the ebonite plate C. A short
piece of indiarubber tubing T was passed over the glass
tube and a projecting flange in which the rod R was screwed.
This served the same purpose as the usual stuffingbox, and
allowed the distance of AB from the uranium to be adjusted
under low pressures.
The plate A was connected to one pole of a battery of
60 volts, the other pole of which was to earth. The plate B
was connected through the screw to one pair of quadrants of
an electrometer, the other pair of which was to earth. In
order to avoid the collection of an electrostatic charge on the
glass surface due to the conduction between the uranium and
the glass near it, it was found very necessary to coat the
inside of the glass cylinder with tinfoil. The tinfoil and
baseplate P were connected to earth.
Since the surface of the uranium layer may be supposed
to be giving out radiation uniformly from all parts, the
intensity of the radiation at points near the centre of the
uranium surface should be approximately uniform. If there
were no absorption of the radiation in the gas, we should
expect the intensity of the radiation to vary but slightly with
the Electrical Conduction produced by it.
12;
distances from the surface sniull compared with the diameter
of the radiating surface.
The radiation passing through the aluminium produces
conductivity between A and B (fig. 3), and the rate of
leak depends on the intensity of the radiation which has
passed through a certain thickness of gas and the aluminium
foil. As the system AB is moved from the baseplate, if
there is a rapid absorption of the radiation in the gas, we
should expect the rate of leak to fall off rapidly, and this
is found to be the case. The following table gives the results
obtained for air, hydrogen, carbonic acid, and coalgas. For
the first reading the distance d of the aluminium foil from the
baseplate was about 3*5 mm.
Kate of leak between plate:
Distance of Al. foil
from Uranium.
Hydrogen.
Air.
Carbonic
Acid.
Coalgas.
d
d + 125 mm.
., + 25 „
„ + 375 „
, + 5
,. + 75 „
„ +10
„ +125 „
„ +15
1
•84
•67
•53
1
'67
•45
•31
•21
•16
1
•74
•57
•41
•32
1
•81
•63
•39
•22
The rate of leak for the distance d is taken as unity in
each gas for the purpose of comparison. The actual rates of
leak between A and B for the distance d is given in the
following table : —
Gas.
JRate of leak
in scaledivisions
per min.
Hydrogen
25
35
28
18
Coalgas
Air
Carbonic acid
13UZ
126.. Prof. E. Rutherford on Uranium Radiation and
The results of the previous table are shown graphically in
fig. 4, where the ordinates represent currents and. the abscissae
Fisr. 4.
distances from the baseplate. It will be seen that the
current decreases most rapidly in carbonic acid and least in
hydrogen. As the distance from the baseplate increases in
arithmetical progression, the rate of leak diminishes approxi
mately in geometrical progression. The rapid decrease of
the current is due to the absorption of the radiation in its
passage through the gas. The decrease of the current in air
at 190 mm. pressure is also shown in the figure. Since the
absorption is smaller for air at this pressure than at normal
pressure, the rate of leak diminishes much more slowly with
the distance.
In the above experiments both the a and ft radiations
produce conductivity in the gas. A thin layer of uranium
oxide was, however, nsed, and in that case the rate of leak due
to the j3 radiation may be neglected in comparison with that
produced by the a radiation.
The results that have been obtained on the variation of the
rate of leak with distance may be simply interpreted on the
theory of the ionization of the gas through which the radia
tion passes. It is assumed that the rate of ionization is
the Electrical Conduction produced by it. 127
proportional to the intensity of the radiation (as is the case
in Rontgenray conduction), and that the intensity of the
radiation near the uranium surface is constant over a plane
parallel to that surface. This is very approximately the case
if the distance from the uranium surface is small compared
with the diameter of the radiating surface.
For simplicity we will consider the case of an infinite
plane of uranium giving out homogeneous radiation.
If I be the intensity of the radiation close to the uranium
surface, the intensity at a distance x is equal to le~ kx where \
is the coefficient of absorption of the gas. The intensity is
diminished in passing through the layer of aluminium foil A
(fig. 3) in a constant ratio for all distances from the uranium.
The intensity at a distance x after passing through the
aluminium is thus Kle~ Kx where a: is a constant. The rate of
production of the ions between two parallel planes between
A and B (fig. 3) at distances x + dx and x from the uranium
is therefore proportional to /cle~ xx dx. If r be the distance of
A from the uranium, and / the distance between A and B,
the total number of ions produced per second between A and
B is proportional to
l+r
I/ce~ kx dx,
I
or to
^T^{le**}.
When a " saturating " electromotive force (see § 16) acts
between A and B, the current is proportional to the total
number of ions produced. Now, as the system AB is moved
fcT.
from the radiating surface, — (1 — e~ xl ) is a constant for any
particular gas. We thus see that the rate of leak is propor
tional to e~ kr , or the rate of leak decreases in geometrical
progression as the distance r increases in arithmetical pro
gression.
This result allows us to at once deduce the value of the
coefficient of absorption for different gases from the data we
have previously given.
The results are given in the following table : —
128 Prof. E. Rutherford on Uranium Radiation and
Gas.
Value of X.
'43
16
23
•93
Carbonic acid
Coalgas
or, to express the same results in a different way, the
intensity of the radiation from an infinite plane of uranium
is reduced by absorption to half its value after having passed
through
3 mm. of carbonic acid,
4' 3 mm. of air ;
7" 5 mm. of coalgas,
16'3 mm. of hydrogen.
We see that the absorption is least in hydrogen and greatest
in carbonic acid, and follows the same order as the density of
the gases.
The values given above are for the <x radiation. The fi
radiation is not nearly so rapidly absorbed as the a, but, on
account of the small electrical leakage produced in its passage
through the gas, it was not found feasible to measure the
absorption in air or other gases ;
The absorption of the a radiation by gases is very much
greater than the absorption of rays from an ordinary Crookes'
tube. In a previous paper * it has been shown that the value
of X for the radiation from the particular bulb used was "01.
The absorption coefficient for the u radiation is 1'6, or 160
times as creat. The absorption of the radiation in gases is
probably of the same order as the absorption for ordinary
#rays.
§ 9. Variation of Absorption with Pressure.
The absorption of the a radiation increases with increase
of pressure and very approximately varies directly as the
pressure.
The same apparatus was used as in fig. 3, and the vessel
was kept connected to an airpump. The variation of the
* Phil. Mag, April 1897.
the Electrical Conduction produced by it.
129
rate of leak between A and B for different distances from the
baseplate was determined for pressures of 760, 370, and
190 mm., and the results are given below : — ■
Rate of leak between plates.
Distance of A
from Uranium.
Air 760 mm.
Air 370 mm.
Air 190 mm.
cl {— 35 mm.)
,, + 2'5 mm.
„ + 5 „
„ + 75 „
„ +10 „
„ +125 „
„ +15 „
1
67
•45
•31
•21
•16
...
1
•71
•51
•36
1
•78
•59
For the purpose of comparison the rate of leak at the
distance d is taken as unity in each case. It can readily be
deduced from the results that the intensity of the radiation is
reduced to half its value after passing through
4'3 mm. c
>f air at 760 mm.
10 „
,. 370 „
195 „
„ 190 „
The absorption is thus approximately proportional to the
pressure for the range that has been tried. It was not
found feasible to measure the absorption at lower pressures
on account of the large distances through which the radiation
must pass to be appreciably absorbed,
A second method of measuring the absorption of the radia
tion in gases, which depends on the variation of the rate
of leak between two plates as the distance between them is
varied, is given in § 12.
§ 10. Effect of Pressure on the Rate of Discharge.
Becquerel * has given a few results for the effects of
pressure, and showed that the rate of leak due to uranium
diminished with the pressure. Beattie and S. de Sinolan f also
* Comptes Rendzis, p. 438 (1897).
t Phil. Mag. xliii. p. 418 (1897).
Phil. Mag, 8. 5. Vol. 47. No. 284. Jan. 1*99.
K
130 Prof. E. Rutherford on Uranium Radiation and
investigated the subject, and came to the conclusion that in
some cases the rate of leak varied as the pressure, and in
others as the square root of the pressure, according to the
voltage employed. Their tabulated results, however, do not
show any close agreement with either law, and in fact, as I
hope to show later, the relation between the rate of leak and
the pressure is a very variable one, depending to a large
extent on the distances between the uranium and the sur
Fig. 5.
rounding conductors, as well as on the gas employed. The
subject is greatly complicated by the rapid absorption of the
radiation by gases, but all the results obtained may be inter
preted on the assumption that the rate of production of ions
at any point varies directly as the intensity of the radiation
and the pressure of the gas.
To determine the effects of pressure, an apparatus similar
to fig. 3 was used, with the difference that the plate _ A was
removed. The uranium compound was spread uniformly
over the central part of the lower plate. The movable
plate ? which was connected with the electrometer, was 10 cm,
the Electrical Conduction produced by it.
131
in diameter and moved parallel to the uranium surface.
The baseplate was connected to one pole of a battery of
100 volts, the other pole of which was connected to earth.
The rate of movement of the electrometerneedle was taken
as a measure of the current between the plates. In some
cases the uranium compound was covered with a thin layer of
aluminium foil, but although this diminished the rate of leak
the general relations obtained were unaltered.
The following tables give the results obtained for air,
hydrogen, and carbonic acid at different pressures with a
potentialdifference of 100 volts between the plates — an
amount sufficient to approximately " saturate " the gases air
and hydrogen. Much larger voltages are required to produce
approximate saturation for carbonic acid.
Air : Uranium oxide on baseplate. Plates about 3*5 mm.
apart.
Air.
Pressure.
Current.
mm.
760
1
600
•86
480
•74
365
•56
210
•32
150
•23
100
•17
50
•088
35
•062
For hydrogen and carbonic acid. Plates about 3*5 mm.
apart.
Hydrogen. Carbonic Acid.
Pressure.
Current.
mm.
760
1
540
•73
335
•46
220
•29
135
•18
Pressure.
Current.
mm.
760
1
410
•92
220
•69
125
•38
55
•175
K2
132 Prof. E. Rutherford on Uranium Radiation and
The current at atmospheric pressure is in each case taken as
unity for comparison, although the actual rates of leak were
different for the three gases. Fig. 5 (p. 130) shows these
results graphically, where the ordinates represent current
and the abscissas pressure. The dotted line shows the position
of the curve if the rate of leak varied directly as the pressure.
It will be observed that for all three gases the rate of leak
first of all increases directly as the pressure, and then
increases more slowly as the pressure increases. The differ
ence is least marked in hydrogen and most marked in
carbonic acid. In hydrogen the rate of leak is nearly pro
portional to the pressure.
The relation between the rate of leak and the pressure
depends also on the distance between the plates. The following
few numbers are typical of the results obtained. There was
a potentialdifference of 200 volts between the plates and the
rate of leak is given in scaledivisions per mm.
Pressure.
Rate of Leak.
Distance between
plates 2*5 mm.
Distance between
plates 15 mm.
mm.
187
376
752
1
11
21
41
47
83
1,7 i
For small distances between the plates the rate of leak is
more nearly proportional to the pressure than for large dis
tances.
The differences between the results for various gases and
for different distances receive a simple explanation if we
consider that the intensity of the radiation falls off rapidly
between the plates on account of the absorption in the gas.
The tables given for the relation between current and pres
sure, where the distance between the plates is small, show
that when the absorption is small, the rate of leak varies
directly as the pressure. For small absorption the intensity
of the radiation is approximately uniform between the plates,
and therefore the ionization of the gas is uniform throughout
the volume of the gas between the plates. Since under a
saturating electromotive force the rate of leak is proportional
the Electrical Conduction produced by it. 133
to the total ionization, the above experiments show that the
rate of production of the ions at any point is proportional to
the pressure. It has been previously shown that the absorp
tion of the radiation is approximately proportional to the
pressure.
Let q = rnte of production of the ions near the uranium
surface for unit pressure.
A = coefficient of absorption of the gas for unit pres
sure.
The total number of ions produced between the plates, distant
d apart, per unit area of the plate is therefore easily seen to
be equal to
f
pq \ eP*** da?;
or to
U 1 
p\ Q d \
P
since we have shown that the ionization and absorption are
proportional to the pressure. If there is a saturating electro
motive force acting on the gas, the ratio of the rate of leak at
the pressure p 1 to that at the pressure p 2 is equal to the ratio r
of the total number of ions produced at the pressure p l to the
total number at pressure p 2 and is given by
1 — e~ p ^o d
Now pJ^Q is the coefficient of absorption of the gas for the
pressure p lt If the absorption is small between the plates,
p{k d and p^d are both small and the value of r reduces to
rts Pi
P?
or the rate of leak when the pressure is small is proportional
to the pressure.
If the absorption is large between the plates at both the
pressures pi and p 2 , the value of r is nearly unity — i. <?. the
rate of leak is approximately independent of the pressure.
Experimental results on this point are shown graphically in
fig. 7 (p. 138). _
For intermediate values of the absorption, the value of r
changes more slowly than the pressure.
With the same distance between the plates, the difference
134 Prof. E. Rutherford on M
ramum
Radiation and
between the curves (fig. 5) for air and hydrogen is due to the
greater absorption of the radiation by the air. The less the
absorption of the gas, the more nearly is the rate of leak pro
portional to the pressure. For carbonic acid the rate of
leak decreases far more slowly with the pressure than for
hydrogen; this is due partly to the much greater value of the
absorption in carbonic acid and partly to the fact that
100 volts between the plates was not sufficient to saturate the
gas.
If we take the rate of leak between two parallel plates
some distance from the source of radiation, we obtain the
somewhat surprising result that the rate of leak increases at
first with diminution of pressure, although a saturating elec
tromotive force is applied.
The arrangement used was very similar to that in fig. 3.
The rate of leak was taken between the plates A and B, which
were 2 cm. apart, and the plate A was about 1*5 cm. from
the uranium surface. The following table gives the results
obtained : —
Pressure.
Current.
mm.
760
1
645
145
525
2
380
22
295
205
180
16
100
104
49
•58
The current at atmospheric pressure is taken as unity.
The results are represented graphically in Rg. 6.
The rate of leak reaches a maximum at a pressure of less
than half an atmosphere, and then decreases, and at a pressure
of 100 mm. the rate of leak is still greater than at atmo
spheric pressure.
This result is readily explained by the great absorption of
the radiation at atmospheric pressure and the diminution of
absorption with pressure.
Let d t = distance of plate A from the uranium.
^2— V V B „
the Electrical Conduction produced by it.
135
With the notation previously used, the total ionization between
A and B (on the assumption that the radiating surface is
infinite in extent) is readily seen to be equal to
i < gMo^l £Mo^2 >
This is a function of the pressure, and is a maximum when
djerMj —d^e~ pK ^ = 0,
i. e. when
The value of pX for air at 760 mm. is 1*6.
Fig. 6.
£00
600
600
If ^2 = 3 cm., ^i = l, the leak is a maximum when the
pressure is about J of an atmosphere. On account of the
large distance of the plates from the uranium surface in
the experimental arrangements, no comparison between
experiment and theory could be made.
In all the investigations on the relation between the pres
sure and the rate of leak, large electromotive forces have
been used to ensure that the current through the gas is
proportional to the total ionization of the gas. With low
voltages the relation between current and pressure would be
136 Prof. E. Rutherford on Uranium Radiation and
very different, and would vary greatly with the voltage and
distance between the electrodes as well as with the gas. It
has not been considered necessary to introduce the results
obtained for small voltages in this paper, as they are very
variable under varying conditions. Although they may all be
simply explained on the results obtained for the saturating
electromotive forces they do not admit of simple calculation,
and only serve to obscure the simple laws which govern the
relations between ionization, absorption, and pressure. The
general nature of the results for low voltages can be deduced
from a consideration of the results given for the connexion
(see § 16) between the current through the gas and the
electromotive force acting on it at various pressures.
The above results for the relation between current and
pressure may be compared with those obtained for Rontgen
radiation. Perrin * found that the rate of leak varied directly
as the pressure for saturating electromotive forces when the
radiation did not impinge on the surface of the metal plates.
This is in agreement with the results obtained for uranium
radiation, for Perrin's result practically asserts that the ioni
zation is proportional to the pressure. The results, however,
of other experimenters on the subject are very variable and
contradictory, due chiefly to the fact that in some cases the
results were obtained for nonsaturating electromotive forces,
while, in addition, the surface ionization at the electrodes
greatly complicated the relation, especially at low pressures.
§11. Amount of Ionization in Different Gases.
It has been shown that the oc radiation from uranium is
rapidly absorbed by air and other gases. In consequence of
this the total amount of ionization produced, when the radia
tion is completely absorbed, can be determined.
The following arrangement was used : — A brass ball 2*2 cm.
in diameter was covered with a thin layer of uranium oxide.
A thin brass rod was screwed into it and the sphere was fixed
centrally inside a belljar of 13 cm. diameter, the brass rod
passing through an ebonite stopper. The belljar was fixed
to a baseplate, and was made airtight. The inside and out
side of the belljar were covered with tinfoil. In practice an
E.M.F. of 800 volts was applied to the outside of the bell
jar. The sphere, through the metal rod, was connected to
one pair of quadrants of an electrometer. It was assumed
that, with such a large potentialdifference between the bell
jar and the sphere, the gas was approximately saturated and
* Comptes Rendus, cxxiii. p. 878.
the Electrical Conduction produced by it.
137
the rate of movement of the electrometerneedle was pro
portional to the total number of ions produced in the gas.
The following were some of the results obtained, the rate of
leak due to air being taken as 100.
Gas.
Total
Ionization.
Air
100
95
106
96
111
102
101
H vdroffen
1 g
Hydrochloric Acid Gas ...
The results for hydrochloric acid and ammonia are only
approximate, for it was found that both gases slightly altered
the radiation emitted by the uranium oxide. For example,
before the introduction of the gas the rate of leak due to air
was found to be 100 divisions in 69 sec; after the introduction
of hydrochloric acid 100 divisions in 72 sec. ; and with air
again after the gas was removed 100 divisions in 74 sec.
The rate of leak is greatest in coalgas and least in hydro
gen, but all the gases tried show roughly the same amount
of ionization as air. In the case considered both kinds of
radiation emitted by uranium are producing ionization in
the gas. By covering over the uranium oxide with a few
layers of thin tinfoil it was found that, for the arrangement
used, the rate of leak due to the penetrating ray was small in
comparison with the rate of leak due to the a, radiation.
The effect of diminution of the pressure on the rate of leak
for air, hydrogen, and carbonic acid is shown in fig. 7,
where the abscissae represent pressure and the ordinates rate
of leak. In the case of air and carbonic acid it was found
that the rate of leak slightly increased at first with diminu
tion of pressure. This was ascribed to the fact that even
with 800 volts acting between the uranium and the surround
ing conductor the saturation for atmospheric pressure was
not complete. It will be observed that the rate of leak in
air remains practically constant down to a pressure of
400 mm., and for carbonic acid down to a pressure of
200 mm. In hydrogen, however, the change of rate of leak
with pressure is more rapid, and shows that all the radiation
138 Prof. E. Rutherford on Uranium Radiation and
emitted by the uranium was not completely absorbed at
atmospheric pressure, so that the total ionization is pro
bably larger than the value given in the table.
Fig. 7.
®
g)
$/
i
1
»
b c?
v /
Jl
/ v/
7
.<?/
1
A
1 (
1 1
U
F
&SSSC//
?£ //v
MMS .
eoo
400
600
800
Assuming that there is the same energy of radiation
emitted whatever the gas surrounding the uranium and that
the radiation is almost completely absorbed in the gas, we
see that there is approximately the same amount of ionization
in all the gases for the same absorption of energy. This is a
very interesting result, as it affords us some information on
the subject of the relative amounts of energy required to
produce ionization in different gases. In whatever process
ionization may consist there is energy absorbed, and the
energy required to produce a separation of the same quantity
of electricity (which is carried by the ions of the gas) is
approximately the same in all the gases tried.
From the results we have just given, it will be seen how
indefinite it is to speak of the conductivity of a gas produced
by uranium radiation. The ratio of the conductivities for
different gases will depend very largely on the distance
apart of the electrodes between which the rate of leak is
observed. When the distance between the electrodes (<?. g.
two parallel plates) is small, the rate of leak is greater in
the Electrical Conduction produced by it. 139
carbonic acid than in air, and greater in air than in hydrogen.
As the distance between the plates is increased, these values
tend to approximate equality. If, however, the rate of leak is
taken between two plates some distance from the radiating
surface (e. g. the plates A and B in fig. 3) , the ratio of the
rates of leak for different gases will depend on the distance
of the plate A from the surface of the uranium. If the
plate A is several centimetres distant from the uranium, the
rate of leak will be greater with hydrogen than with air, and
greater in air than in carbonic acid — the exact reverse of the
other case. These considerations will also apply to what is
meant by the conductivity of a gas for uranium radiation.
In a previous paper * I found the coefficient of absorption
of a gas for Rontgen rays to be roughly proportional to the
conductivity of the gas. The conductivity in this case was
measured by the rate of leak between two plates close together
and not far from the Crookes tube. The absorption in the
air between the bulb and the testing apparatus was small.
If it were possible to completely absorb the Rontgen radia
tion in a gas and measure the resulting conductivity, the total
current should be independent of the gas in which the radia
tion was absorbed. This result follows at once if the ab
sorption is proportional to the ionization produced for all
gases. The results for uranium and Rontgen radiation are
thus very similar in this respect.
§ 12. Variation of the Current between two Plates with the
Distance behveen them.
The experimental arrangement adopted was similar to that
in fig. 3 with the plate A removed. Two horizontal polished
zinc plates 10 cm. in diameter were placed inside a belljar.
The lower plate was fixed and covered with a uniform layer
of uranium oxide, and the upper plate was movable, by
means of a screw, parallel to the lower plate. The belljar
was airtight, and was connected with an airpump. The
lower plate was connected to one pole of a battery of 200
volts, the other pole of which was earthed, and the insulated
top plate was connected with the electrometer. The exterior
surface of the glass was covered with tinfoil connected to
earth.
The following table gives the results of the variation of the
rate of leak with distance for air at pressures of 752, 376,
and 187 mm. The results have been corrected for change of
* Phil. Mag. April 1897.
140 Prof. E. Rutherford on Uranium Radiation and
the capacity of the electrometer circuit with movement of the
plates.
Distance between
plates.
Hate of leak in scaledivisions per min.
752 mm.
376 mm. 187 mm.
mm.
25
41
21
5
70
40
20
75
92
53
10
109
65 36
125
123
76
15
128
83
47
The results are shown graphically in fig. 8, where the
Fie. 8.
abscissae represent distances between the plates and the ordi
nates rates of leak. The values given above correspond to
saturation rates of leak ; for 200 volts between the plates is
the Electrical Conduction produced by it. 141
sufficient to very approximately saturate the gas even for the
greatest distance apart of 1*5 cm.
It will be observed that the rate of leak increases nearly
proportionally to the distance between the plates for short
distances, but for air at atmospheric pressure increases very
slowly with the distance when the distances are large.
If there were no appreciable absorption of the radiation by
the gas, the ionization would be approximately uniform
between the plates, provided the diameter of the uranium
surface was large compared with the greatest distance between
the plates. The saturation rate of leak would in that case
vary as the distance. If there is a large absorption of the
radiation by the gas. the ionization will be greatest near the
uranium and will fall off rapidly with the distance. The
saturation rate of leak will thus increase at first with the
distance, and then tend to a constant value when the radiation
is completely absorbed between the plates.
The results given in the previous table allow us to deter
mine the absorption coefficient of air at various pressures.
My attention was first drawn to the rapid absorption of the
radiation by experiments of this kind.
The number of ions produced between two parallel plates
distant d apart is equal to
Jo
(1e
pq
i. e., to
I
assuming the ionization and the absorption are proportional
to the pressure. The notation is the same as that used
in § 10.
For the pressure p the saturation rate of leak between the
plates is thus proportional to 1 — e~ pk ° d .
If p and d are varied so that p x d is a constant, the rate
of leak should be a constant. This is approximately true as
the numbers previously given (see fig. 8) show. It must,
however, be borne in mind that the conditions, on which the
calculations are based, are only approximately fulfilled in
practice, for we have assumed the uranium surface to be
infinite in extent and that the saturation is complete.
The variation of the rate of leak with distance agrees fairly
closely with the theory. When p\ d is small the rate of leak
is nearly proportional to the distance between the plates and
the pressure of the gas. When p\ d is large the rate of leak
varies very slowly with the distance,
142 Prof. E. Rutherford on Uranium Radiation and
The value of p\ can be deduced from the experi
mental results, so that we have here an independent method
of determining the absorption of the radiation at different
pressures.
The lower the pressure the more uniform is the ionization
between the plates, so that the saturation rate of leak at low
pressures is nearly proportional to the distance between the
plates. This is seen to be the case in fig. 8, where the curve
for a pressure of 187 mm. is approximately a straight line.
Similar results have been obtained for hydrogen and carbonic
acid.
§ 13. Rate of Recombination of the Ions.
Air that has been blown by the surface of a uranium com
pound has the power of discharging both positive and negative
electrification. The following arrangement was used to find
the duration of the afterconductivity induced by uranium
radiation : — A sheet of thick paper was covered over with a
thin layer of gumarabic, and then uranium oxide or uranium
potassium sulphate in the form of fine powder was sprinkled
over it. After this had dried the sheet of paper was formed
into a cylinder with the uranium layer inside. This was then
placed in a metal tube T (fig. 9) of 4 cm. diameter. A
Fig. 9.
Earth
blast of air from a gasometer, after passing through a plug C
of cottonwool to remove dust, passed through the cylinder T
and then down a long metal tube connected to earth.
Insulated electrodes A and B were fixed in the metal tube.
The electrometer could be connected to either of the elec
trodes A or B. In practice the quadrants of the electro
meter were first connected together. The electrode A or B
and the electrometer were then charged up to a potential of
30 volts, and the quadrants then separated.
When the uranium was removed there was no rate of leak
at either A or B when a rapid current of air was sent through
the tube. On replacing the uranium cylinder and sending a
current of air along the tube, the electrometer showed a
the Electrical Conduction produced by it. 143
gradual loss of charge which continued until the electrode
was discharged.
When the electrode A was charged to 30 volts there was
no rate of leak of B. The rate of leak of B or A is thus
proportional to total number of ions in the gas. The ions
recombine in the interval taken for the air to pass between
A and B. The rate of leak of B for a saturating voltage,
when A is to earth, is thus less than that of A.
For a particular experiment the rate of leak of the electrode
A was 146 divisions per minute. When A was connected to
earth, the saturation rate of leak of B was 100 divisions per
minute. The distance between A and B was 44 cm., and the
mean velocity of the current of air along the tube 70 cm. per
second. In the time, therefore, of *63 sec. the conductivity
of the gas has fallen to '68 of its value.
If we assume, as in the case of Bontgenized air *, that the
loss of conductivity is due to the recombination of the ions,
the variation of the number with the time is given by
dn „
dt=« n >
where n is the number of ions per c.c. and a a constant. If
N is the number of ions at the electrode A, the number of
ions n at B after an interval t is given by
n N
JSTow the saturating rates of leak at A and B are propor
tional to N and n, and it can readily be deduced that the
time taken for the number of ions to recombine to half their
number is equal to 1*3 sec. This is a much slower rate of
recombination than with Bontgenized air near an ordinary
Crookes tube.
The amount of ionization by the uranium radiation is in
general much smaller than that due to Bontgen rays, so that
the time taken for the ions to fall to half their number is
longer.
The phenomenon of recombination of the ions is very similar
in both uranium and Bontgen conduction. In order to test
whether the rate of recombination of the ions is proportional
to the square of the number present in the gas, the following
experiment was performed : —
* Phil. Mag. Nov, 1897,
144 Prof. E. Rutherford on Uranium Radiation and
A tube A (fig. 10) was taken, 3 metres long and 5" 5 cm.
in diameter. A cylinder D, 25 cm. long, had its interior
surface coated with uranium oxide. This cylinder just fitted
the large tube, and its position in the tube could be varied
by means of strings attached to it, which passed through
corks at the ends of the long tube. The air was forced
•Fiff. 10.
D __ W&
Earth
Earth
through the tube from a gasometer, and on entering the tube
A passed through a plug of cottonwool, E, in order to remove
dust from the air and to make the current of air more
uniform over the crosssection of the tube. The air passed
by the uranium surface and then through a gauze L into
the testing cylinder B of 2*8 cm. diameter. An insulated
rod C, 1*6 cm. in diameter, passed centrally through the
cylinder B and was connected with the electrometer. The
cylinders A and B were connected to one pole of a battery of
32 volts, the other pole of which was to earth.
The potentialdifference of 32 volts between B and C was
sufficient to almost completely remove all the ions from the
gas in their passage along the cylinder. The rate of leak of
the electrometer was thus proportional to the number of ions
in the gas.
The following rates of leak were obtained for different
distances of the uranium cylinder from the gauze L
(table, p. 145).
The first column of the table gives the distances of the
end of the uranium cylinder from the gauze L. d (about
20 cm.) was the distance for which the first measurement
was made. In the second column the time intervals taken
for the air to pass over the various distances are given. The
value of t corresponds to the distance d. The mean velocity
of the current of air along .the tube was about 25 cm. per sec.
the Electrical Conduction produced by it.
145
Di stance of
Uranium cylinder
from L.
T.
Kate of leak in
scaledivisions
per minute.
Calculated
rates of leak.
d
d+25 cm.
cZ+50 „
d+100 „
d+200 „
t
t\\ sec.
*+2 „
*+4 „
'+8 „
*159
111
* 87
62
39
*159
112
* 87
60
37
In the third column are given the observed rates of leak, and
in the fourth column the calculated values.
The values were calculated on the assumption that the rate
of recombination of the ions was proportional to the square
of the number present, i. e. that
dn
dt
— an'
where n is the number of ions present and a is a constant.
The two numbers with the asterisk were used to determine
the constants of the equation. The agreement of the other
numbers is closer than would be expected, for in practice the
velocity of the blast is not constant over the crosssection,
and there is also a slight loss of conductivity of the gas due
to the diffusion of the ions to the side of the long tube.
It will be observed that the rate of recombination is very
slow when a small number of ions are present in the gas, and
that the air preserves one quarter of its conducting power
after an interval of 8 seconds.
§ 14. Velocity of the Ions.
The method* adopted to determine the velocity of the
ions in Eontgen conduction cannot be employed for uranium
conduction. It is not practicable to measure the rate of re
combination of the ions between the plates on account of the
very small afterconductivity in such a case ; and, moreover,
the inequality of the ionization between the two plates greatly
disturbs the electric field between the plates.
A comparison of the velocities, under similar conditions, of
the ions in Eontgen and uranium conduction can, however,
Phil. Mag. S.
* Phil. Mag. Nov. 1897.
5. Vol. 47. No. 284. Jan.
1899.
L
146 Prof. E. Butherford on Uranium Radiation and
be readily made. The results stow that the ions in the two
types of conduction are the same.
In order to compare the velocities an apparatus similar to
fig. 10 was used. The ions were blown by a charged wire A,
and the conductivity of the gas tested immediately afterwards
at an electrode B, which was fixed close to A. The electrode
A was cylindrical and fixed centrally in the metal tube L,
which was connected to earth. For convenience of calcula
tion it is assumed that the electric field between the cylinders
is the same as if the cylinders were infinitely long.
Let a, b be the radii of the electrode A and the tube L
(internal) ;
Let V be the potential of A (supposed positive).
The electromotive intensity X (without regard to sign) at
a distance r from the centre of the tube is given by
X=
i b
r lofif 
Let u l u 2 be the velocities of the positive and negative ions
for a potential gradient of 1 volt per cm. If the velocity is
proportional to the electric force at any point, the distance dr
traversed by the negative ion in the time dt is given by
dr = X.u%dt,
log e rdr
or dt = — ^f .
\u 2
Let r 2 be the distance from the centre from which the
negative ion can just reach the electrode in the time t taken
for the air to pass along the electrode.
Then
£ = MU^ — ; log.
2V?
&e
a
If p 2 be the ratio of the number of the negative ions that
reach the electrode A to the total number passing by,
Ta — a
then p 2 = '
Therefore
Va 2
p 2 {b*a*) log
—  (1)
2 V
the Electrical Conduction produced by it. 147
Similarly the ratio p x of the number of positive ions that
give up their charge to the external cylinder to the total
number is given by
lh = W7t — (2)
In the above equations it is assumed that the current of
air is uniform over the crosssection of the tube, and that the
ions are uniformly distributed over the crosssection ; also,
that the movement of the ions does not appreciably disturb
the electric field. Since the value of t ran be calculated from
the velocity of the current of air and the length of the elec
trode, the values of the velocities of the ions under unit
potential gradient can at once be determined.
The equation (1) shows that p 2 is proportional to V, —
i. e. that the rate of leak of the electrode A varies directly as
the potential of A, provided the value of V is not large
enough to remove all the ions from the gas as it passes
by the electrode. This was experimentally found to be the
case.
In the comparison of the velocities the potential V was
adjusted to such a value that p 2 was about one half. This was
determined by testing the rate of leak at B with a saturating
electromotive force. The amount of recombination of the
ions between the electrodes A and B was very small, and
could be neglected.
The uranium cylinder was then removed, all the other
parts of the apparatus remaining unchanged. An aluminium
cylinder was substituted for the uranium cylinder, and ,^rays
were allowed to fall on the aluminium. The bulb and
inductioncoil were placed in a metal box in order to care
fully screen off all electrostatic disturbances. The rays were
only allowed to fall on the central portion of the cylinder.
The intensity of the rays was adjusted so that, with the same
current of air, the rate of leak was comparable with that
produced by the uranium. It was then found that the value
of p2 was nearly the same as for the uranium conduction.
For example, the rate of leak of B was reduced from 38 to
14 scaledivisions per min. by charging A to a certain small
potential, when the air was blown by the surface of the
uranium. When Rontgenized air was substituted, the rate of
leak was reduced from 50 to 18 divisions per min. under the
same conditions. The values of p 2 were '63 and *64 respec
tively. This agreement is closer than would be expected,
as the bulb was not a very steady source of radiation.
L 2
148 Prof. E. Rutherford on Uranium Radiation and
This result shows that the ions in Eontgen and uranium
conduction move with the same velocity and are probably
identical. The velocity of an ion in passing through a gas
is proportional to — , where e is the charge carried by the ion,
and m its mass. Unless e and m vary in the same ratio it
follows that the charge carried by the ion in uranium and
Eontgen conduction is the same, and also that their masses
are equal.
It was found that the velocity of the negative ion was
somewhat greater than that of the positive ion. This has
been shown to be the case for ions produced by Eontgen
rays *. The difference of velocity between the positive and
negative carrier is readily shown. The rate of leak of B is
observed when charged positively and negatively. When B
was charged positively the rate of leak measured the number
of negative ions that escaped the electrode A, and when
charged negatively the number of positive ions. The rate of
leak was always found to be slightly greater when B was
charged negatively. This is true whether A is charged
positively or negatively, and shows that there is an excess of
positive ions in the gas after passing by the electrode A.
The difference of velocities of the ions can also readily
be shown by applying an alternating electromotive force to
the electrode A sufficient to remove a large proportion of the
ions as the air passes by. The issuing gas is always found
to be positively charged, showing that there is an excess of
positive over negative ions.
A large number of determinations of the velocities of the
uranium ions have been made, with steady and alternating
electromotive forces, w T hen the air passed between concentric
cylinders or plane rectangular plates. In consequence of the
inequality of the velocity of the current of air over the
crosssection of the tube, and other disturbing factors which
could not be allowed for, the determination could not be
made with the accuracy that was desired. For an accurate
determination, a method independent of currents of air is
very desirable.
§ 15. .Potential Gradient between two Plates.
The normal potential gradient between two plates is altered
by the movement of the ions in the electric field.
Two methods were used to determine the potential gradient.
* Zeleny, Phil. Mag. July 1898.
the Electrical Conduction produced by it. 149
In the first method a thin wire or strip was placed between
two parallel plates one of which was covered with uranium.
The wire was connected with the electrometer, and after being
left some time took up the potential in the air close to the
wire. In the second method the ordinary mercury or
waterdropper was employed to measure the potential at a
point.
For the first method two large zinc plates were taken
and placed horizontal and parallel to one another. A layer
of uranium oxide was spread over the lower plate. The
bottom plate was connected to one pole of a battery of 8 volts,
and the top plate was connected to earth. An insulated thin
zinc strip was placed between the plates and parallel to
them. The strip was connected with the electrometer, and
gradually took up the potential of the point. By moving
the strip the potential at different points between the plates
could be determined.
The following table is an example of the results obtained.
Plates 4' 8 cm. apart ; 8 volts between plates.
Distance from
top plate.
Potential in
volts
with Uranium.
Potential in volts
without
Uranium.
•6
12
21
31
48
25
38
59
7
8
1
2
35
52
8
The third column is calculated on the assumption that without
the uranium the potential falls off uniformly between the
plates.
The method given above is not very satisfactory when the
strip is close to the plates, as it takes up the potential of the
point very slowly.
The water or mercurydropper was more rapid in its
action, and gave results very similar to those obtained by the
first method. Two parallel brass plates were placed vertically
and insulated. One plate was connected to the positive and
150 Prof. E. Rutherford on Uranium Radiation and
the other to the negative pole of a battery. The middle
point of the battery was placed to earth. The waterdropper
was connected with the electrometer. The potential at a point
was first determined without any uranium near. One plate
was then removed, and an exactly similar plate, covered with
the uranium compound, substituted. The potential of the
point was then observed again. In this way the potential at
any point with and without the uranium could be determined.
The curve shown in fig. 11 is an example of the potential
gradient observed between two parallel plates 6' 6 cm. apart.
The dotted line represents the potential gradient when the
uranium is removed. The ordinates represent volts and the
abscissae distances from the plate covered with the uranium
compound.
Fio«. ll.
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It will be observed that the potential gradient is diminished
near the uranium and increased near the other plate. The
point of zero potential is displaced away from the uranium.
From curves showing the potential gradient between two
plates, the distribution of free electrification between the plates
can be deduced. By Taking the first differential of the curve
we obtain j , the electric force at anv point, and by taking
dx * . d 2 v
the second differential of the curve we obtain j— 2 , which is
the Electrical Conduction produced by it. 151
equal to — ±irp, where p is the volumedensity of electrifica
tion at any point. In order to produce the disturbance of
the electric field shown in fig. 11, there must be an excess
of ions of one kind distributed between the plates. Such a
result follows at once from what has been said in regard to
the inequality of the ionization between the plates due to the
absorption of the radiation.
It was found that the potential gradient approached more
and more its undisturbed value with increase of the electro
motive force between the plates. The displacement of the
point of zeropotential from the uranium surface increased
with diminution of electromotive force. For example, for
two plates 51 mm. apart, charged to equal and opposite
potentials, the points of zero potential were 28, 30, 33 mm.
from the uranium when the differences of potential between
the plates were 16, 8, and 4 volts respectively.
When the uranium was charged positively, the point of
zero potential was more displaced than when it was charged
negatively. This is due to the slower velocity of the positive
ion.
The slope of potential very close to the surface of the
uranium has not been investigated. The deviation from the
normal potential slope between the plates depends very
largely on the intensity of the ionization produced in the gas.
With very weak ionization the normal potential gradient is
only slightly affected.
Child * and Zeleny f have shown that the potential gradient
between two parallel plates exposed to Rontgen rays is not
uniform. In their cases the ionization was uniform between
the plates, and the disturbance in the field manifested itself in
a sudden drop at both electrodes. In the case considered
for uranium radiation, the ionization is too small for this
effect to be appreciable. The disturbance of the field is due
chiefly to the inequality of the ionization, and does not only
take place at the electrodes.
§ 16. Relation between Current and Electromotive Force.
The variation with electromotive force of the current
through a gas exposed to uranium radiation has been investi
gated by Becquerel {, and later by de Smolan and Beattie§.
* Wied. Annal. April 1898, p. 152,
t Phil. Mag., July 1898.
\ Comptes Rendus, pp. 438, 800 (1897).
§ Phil. Mag. vol. xliii. p. 418 (1897).
152 Prof. E. Rutherford on Uranium Radiation and
The general relation between the current through the gas and
the E.M.F. acting on it is very similar to that obtained for
gases exposed to Rontgen radiation. The current at first
increases nearly proportionally with the E.M.F. (provided
the E.M.F/s of contact between the metals are taken into
account), then more slowly, till finally a stage is reached,
which may be termed the " saturation stage," where there is
only a very slight increase of current with a very large
increase of electromotive force. As far as experiments have
gone, uranium oxide, when immersed in gases which do not
attack it, gives out a constant radiation at a definite tempera
ture, and the variation of the intensity of radiation with the
temperature over the ordinary atmospheric range is inap
preciable. For this reason it is possible to do more accurate
work with uranium radiation than with Rontgen radiation,
for it is almost impossible to get a really steady source of
a?rays for any length of time.
It was the object of these experiments to determine the
relation between current and electromotive force with accuracy,
and to see whether the gas really becomes saturated ; i. €.,
whether the current appreciably increases with electromotive
forces when the electromotive forces are great, but still not
sufficient to break down the gas and to produce conduction
in the gas without the uranium radiation.
A null method was devised to measure the current, in order
to be independent of the electrometer as a measuring instru
ment and to merely use it as an indicator of difference of
potential.
Rff. 12.
Ear
E/lffTH
Fig. 12 shows the general arrangement of the experiment.
A and B were two insulated parallel zinc plates : on the lower
the Electrical Conduction produced by it. 153
plate A was spread a uniform layer of uranium oxide. The
bottom plate was connected to one pole of a battery of a large
number of storagecells, the other pole of which was to earth.
The insulated plate B was connected to one pair of quadrants
of an electrometer, the other pair of which was to earth.
Under the influence of the uranium the air between the plates
A and B is made a partial conductor, and the potential of B
tends to become equal to that of A. In order to keep the
potential of B at zero, B is connected through a very high
resistance T of xylol, one end of which is kept at a steady
potential. If the amount of electricity supplied to B through
the xylol by the battery is equal and opposite in sign to the
quantity passing between A and B, the potential of B will
remain steadily at zero. In order to adjust the potential to
be applied to one end of the xyloltube T, a battery was
connected through resistanceboxes Rj R 2 , the wire between
being connected to earth. The ratio of the E.M.F. e acting
on T to the E.M.F. E of the battery is given by
e
Ri
E Bi + R 2 *
In practice Ri + Rg was always kept constant and equal to
10,000 ohms, and, in adjusting the resistance, plugs taken
from one box were transferred to the other. The value of e
is thus proportional to R 1; and the amount of current supplied
to B (assuming xylol obeys Ohm's Law) is proportional to R x .
If the resistances are varied till the electrometer remains at
the " earth zero," the current between the plates is pro
portional to Ri. If the value of the E.M.F. applied is too
great the needle moves in one direction, if too small in the
opposite direction. For fairly rapid leaks the current could
be determined to an accuracy of 1 per cent.; but for slow
electromotive leaks this accuracy is not possible on account
of slow changes of the electrometer zero when the quadrants
are disconnected.
The following tables show the results of an experiment with
uranium oxide. The surface of the uranium was II cm.
square. In order to get rid of stray radiation at the sides
lead strips, which nearly reached to the top plate, were placed
round the uranium. 16 volts were applied to the resistance
box, and a resistance of 10,000 ohms kept steadily in the
circuit.
154 Prof. E. Rutherford on Uranium Radiation and
Plates 2*5 cm. apart.
Plates *5 cm. apart.
Volts.
Current R 1 .
Volts. 1 Current E r
•5
1
2
4
8
16
375
112
375
800
425
825
1570
2750
3750
4230
4700
5250
5625
5825
■125
•25
•5
1
2
4
8
16
100
335
1400
2800
4300
5250
5650
6200
6670
6950
7400
7850
Under the column of volts the difference of potential
between A and B is given. The current is given in terms of
the resistance Ex required to keep the electrometer at the
earth zero. It will be observed that for the first few readings
Ohm's Law is approximately obeyed, and then the current
increases more gradually till for large E.M.F/s the rate of
increase is very slow. For the plates '5 cm. apart the
rate of leak for 335 volts is only 50 per cent, greater than
the rate of leak for 1 volt.
The same general results are obtained if the surface of the
uranium is bare or covered with thin metal. The disadvantages
of covering the surface with thin tin or aluminium foil are (1)
that the intensity of the radiation is considerably decreased ;
(2) that the ions diffuse from under the tinfoil through any
small holes or any slight openings in the side. The drawback
of using the uncovered uranium in the form of fine powder 3
is that under large electric forces the fine uranium particles
are set in motion between the plates and cause an additional
leakage. In practice the rate of leak was measured with
potentialdifferences too small to produce any appreciable
action of this kind.
In order to investigate the currentelectromotiveforce
relations for different gases the same method was used, but
the leakage in this case took place between two concentric
cylinders. The apparatus is shown in the lower part of
fig. 12 : C and D were two concentric cylinders of brass 4*5
and 3*75 cm. in diameter, insulated from each other.
the Electrical Conduction produced by it.
155
The ends of the cylinder D were closed by ebonite collars,
and the central cylinder was supported in position by brass
rods passing through the ebonite. The surface was uniformly
covered with uranium oxide. The cylinder D was connected
to one pole of a battery, the other pole of which was to earth.
The cylinder C was connected to the electrometer. The
following tables show the results obtained for Irydrogen, carbonic
acid, and air. Distance between cylinders *375 cm.
Hydrogen.
Carbonic Acid.
Air.
Volts.
Current.
122
•062
125
•125
123
•25
142
•5
150
1
160
2
163
4
165
8
1C8
16
172
108
178
216
185
Volts.
Current.
95
•125
205
•25
255
•5
305
1
355
2
405
4
460
8
520
16
590
36
705
108
787
216
820
Volts.
Current.
I
+ 1
418
2
451
4
495
8
533
36
601
108
615
216
630
The above results are expressed graphically in fig. 13,
where the ordinates represent current on an arbitrary scale
and the abscissse volts. In the tables given for hydrogen and
carbonic acid it will be observed that the current has a
definite value when there is no external electromotive force
acting. The reason for this is probably due to the contact
difference of potential between the uranium surface and the
interior brass surface of the outside cylinder. When the
external cylinder was connected to earth the inside cylinder
became charged* to —'12 volt after it was left a short time.
* This phenomenon has been studied by Lord Kelvin, Beattie, and
S. de Smolan, and it has been shown that metals are charged up to small
potentials under the influence of uranium radiation. The steady
difference of potential between two metal plates between which the
radiation falls is the same as the contact difference of potential. An
exactly similar phenomenon has been studied by Perrin (Comptes Rendus,
cxxiii. p. 496) for #rays.
156 Prof. E. Rutherford on Uranium Radiation and
In consequence of this action, for small electromotive forces
the rates of leak are different for positive and negative.
Fiar. 13.
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40
80
120
160
200
Results of this kind are shown more clearly in fig. 14, which
gives the currentelectromotiveforce curves for hydrogen
and carbonic acid for small voltages. When there is no
external electromotive force acting, the current has a fixed
value ; if the uranium is charged positively, the current
increases slowly with the voltage ; when the uranium is
charged negatively, the current is at first reversed, becomes
zero, and rapidly increases with the voltage until for about
1 volt between the plates the positive and negative currents
are nearly equal. The curve for carbonic acid with a positive
charge on the uranium is also shown. It will be seen that
the initial slope of the curve is greater for carbonic acid than
for hydrogen.
It is remarkable that the current with zero E.M.F. for
hydrogen is about twothirds of its value when 216 volts are
acting between the plates. The ions in hydrogen diffuse more
rapidly than in air, and in consequence a large proportion
the Electrical Conduction produced by it, 157
of the negative ions reach the uranium and give up their
charge to it before recombination can take place.
Fig. 14.
If the radiation fell between two plates of exactly the same
metal, the inequality between the positive and negative
current values for low voltages would almost disappear, but
even in that case there would still be an apparent current
through the gas, due to the fact that the negative carriers
diffuse with greater rapidity than the positive. Effects of
this kind have been studied for Eontgen radiation by Zeleny *.
For large E.M.F.'s no appreciable difference in the value
of the current could be detected whether the uranium was
positively or negatively charged, i. e. positive and negative
electrifications are discharged with equal facility.
For the different gases the current tends more rapidly to a
saturation value in hydrogen than in air, and more rapidly in
air than in carbonic acid. In all these cases there is still a
slight increase of current with increase of E.M.F. long after
the " knee " of the saturation curve has been passed, and in
no case has complete saturation been observed at atmospheric
pressure, even for a potential gradient of 1300 volts per cm.
The explanation of the general form of the curves showing
the relation between current and electromotive force for
ionized gases has been given in a previous paper f. In the
* Phil. Mag., July 1898.
f J. J. Thomson and E. Rutherford, Phil. Mag. November 1896.
158 Prof. E. Rutherford on Uranium Radiation and
case of uranium conduction the phenomenon is still further
complicated by the want of uniformity of ionization between
the plates and the resulting disturbance of the electrostatic
field due to the excess of ions of one kind between the
plates.
The ionization of the gas is greatest near the uranium sur
face, and falls off rapidly with the distance. The rate of
recombination of the ions thus varies from point to point
between the plates, being greatest near the surface of the
uranium.
The equations which express completely the relation between
the current and electromotive force for the rate of leak between
two parallel plates, one of which is covered with uranium, are
very complex and cannot be expressed in simple form. The
disturbance of the electrostatic field between the plates, due to
the movement of the ions, has to be considered as well as
the variable rates of recombination at the different points, and
the difference of velocity between the positive and negative
ions.
The great difficulty in producing complete saturation, i. e. to
reach a stage when all the ions produced reach the electrodes,
may be due to one or more of three causes: —
(1) Rapid rate of recombination of the ions very near the
surface of the uranium.
(2) Presence of very slow moving ions together with the
more rapidly moving carriers.
(3) An effect of the electric field on the production of
the ions.
The effect of (3) is probably very small, for there is no
experimental evidence of any such action unless the electro
motive forces are very high. That the slow increase of the
current in strong fields is due to (1) rather than (2) receives
some support from an experiment that has been recently
tried. Instead of measuring the current with the uranium
covering one electrode, the air which had passed over uranium
was forced between two concentric cylinders between which
the electromotive force was acting. The rate of leak was
found to only increase 2 or 3 per cent, when the E.M.F. was
increased from 16 to 320 volts. This increase is much smaller
than in the results previously given. Since the effect of (2)
would be present in both cases, this experiment seems to show
that the difficulty in removing all the ions from the gas is not
due to the presence of some very slowmoving carriers.
the Electrical Conduction produced by it.
Effect of Pressure.
159
Some current electromotiveforce curves for small voltages
have been obtained at different pressures. Examples of the
results are shown in fig. 15. which gives the relation between
the current and the electromotive force at pressures of 760,
380, 190, and 95 mm. of mercury.
These results were obtained with a different apparatus and
by a different method to that given in fig. 12. Two parallel
insulated metal plates, about 3 cm. apart, one of which was
covered with uranium oxide, were placed inside an airtight
vessel. One plate was connected to earth and the other to
the electrometer. The plate connected to the electrometer
was then charged up to a potential of 10 volts. On account
of the presence of the uranium oxide the charge slowly leaked
away, and the rate of movement of the electrometerneedle
measured the current corresponding to different values of the
electromotive force.
Fig. 15.
f/
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Air
190 MM
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Air 9
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V
Volt
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The method did not admit of the accuracy of that pre
viously employed (see fig. 12). The rate of leak for small
fractions of a volt could not be determined, so that in the
curves fig. 15 it is assumed that the current was zero when
the electromotive force was zero. This is probably not quite
160 Prof. E. Rutherford on Uranium Radiation and
accurate owing to the slight contactdifference of potential
between the plates, so that there was a small initial current
for zero external electromotive force.
The general results show that the gas tends to become more
readily saturated with diminution of pressure. The variation
of the current with the E.M.F. depends on two factors — the
velocities of the ions, and their rate of recombination. Some
experiments on the velocity of the carriers * in ultraviolet
light conduction showed that the velocity Of the ions in a
given electric field is inversely proportional to the pressure.
This is probably also true for the ions in Rontgen conduction;
so that under the pressure of 95 mm. the ions would move
eight times as fast as at atmospheric pressure. The variation
of the rate of recombination with pressure has not yet been
determined.
The curve for hydrogen at atmospheric pressure is also
given in fig. 15, and shows that hydrogen is about as easily
saturated as air at 190 mm. pressure.
§ 17. Separation of the Positive and Negative Ions.
It is a simple matter to partially separate the positive and
negative ions in uranium conductions and produce an
electrified gas, The subject of the production of electrifi
cation by passing a current of air over the surface of uranium
enclosed in a metal vessel has been examined by Beattie f,
who found the electrification obtained was of the same sign
as the charge on the uranium. His results admit of a simple
explanation on the theory of ionization. The gas near the
surface of the uranium is far more strongly ionized than that
some distance away on account of the rapid absorption of the
radiation by the air. For convenience of explanation, let us
suppose a piece of uranium, charged positively, placed inside
a metal vessel connected to earth, and a current of air passed
through the vessel. Under the influence of the electromotive
force the negative ions travel in towards the uranium, and the
positive ions towards the outer vessel. Since the ionization
is greater near the surface of the uranium, there will be an
excess of positive ions in the air some distance away from
the uranium. Part of this is blown out by the current of air,
and gives up its charge to a filter of cottonwool. The total
number of negative ions blown out in the same time is much
less, as the electromotive intensity, and therefore the velocity
of the carrier, is greater near the uranium than near the out
side cylinder. Consequently there is an excess of positive
* Proc. Oamb. Phil. Soc. Feb. 21, 1888.
f Phil. Mag. July 1897, xliv. p. 102,
the Electrical Conduction produced by it. 161
ions blown out, and a positively electrified gas is obtained.
As the potentialdifference between the electrodes is increased,
the amount of electrification obtained depends on two opposing
actions. The velocity of the carriers is increased, and conse
quently the ratio of the number of carriers removed is dimi
nished. But if the gas is not saturated, with increase of
electromotive force the number of ions travelling between the
electrodes is increased, and for small voltages this increase
more than counterbalances the diminution due to increase of
velocity. The amount of electrification obtained will there
fore increase at first with increase of voltage, reach a maxi
mum, and then diminish ; for when the gas is saturated no
more ions can be supplied with increase of electromotive
force. This is exactly the result which Beattie obtained, and
which I also obtained in the case of the separation of the ions
of Rontgenized air. The fact that more positive than nega
tive electrification is obtained is due to the greater velocity
with which the negative ion travels. (See § 14.)
The properties of this electrified gas are similar to that which
has been found from Rontgen conduction. The opposite sign
of the electrification obtained by Beattie for uranium in
duction, and by myself for Rontgen conduction *, is to be
expected on account of the different methods employed. For
obtaining electrification from Rontgenized air a rapid current
of air was directed close to the charged wire. In that case
the sign of the electrification obtained is opposite to that of
the wire, as it is the carriers of opposite sign to the wire which
are blown out before they reach the wire. In the case of
uranium the current of air filled the crosssection of the space
between the electrodes ; and it has been shown that under such
conditions electrification of the same sign as the uranium is to
be expected.
§ 18. Dischargingpower of Fine Gauzes.
Air blown over the surface of uranium loses all trace of con
ductivity after being forced through cottonwool or through
any finely divided substance. In this respect it is quite
similar to Rontgenized air. The dischargingpower of cotton
wool and fine gauzes is at first sight surprising, for there is
considerable evidence that the ions themselves are of molecular
dimensions, and might therefore be expected to pass through
small orifices ; but a little consideration shows that the ions,
like the molecules, are continually in rapid motion, and, in
addition, have free charges, so that whenever they approach
within a certain distance of a solid body they tend to be
attracted towards it, and give up their charge or adhere to
* Phil. Mag. April 1897.
Phil Mag. S. 5. Vol, 47. No. 284. Jan. 1899. M
162 Uranium Radiation and Electrical Conduction.
the surface. On account of the rapidity of diffusion * of the
ions, the dischargingpower of a metal gauze, with openings
very large compared with the diameter of a carrier, may be
considerable. The table below gives some results obtained for
the dischargingpower of fine copper gauze. The copper
gauze had two strands per millim., and the area occupied by
the metal was roughly equal to the area of the openings. The
gauzes filled the crosssection of the tube at A (fig. 9), and
were tightly pressed together. The conductivity of the air
was tested after its passage through the gauzes, the velocity
of the air along the tube being kept approximately constant.
The rates of leak per minute due to the air after its passage
through different numbers of gauzes is given below.
Number of Gauzes.
Rate of leak in divisions
per minute.
44
325
26 5
195
105
6
1
2
3
4
5
After passing through 5 gauzes the conductivity of the air
has fallen to less than ^ of its original value. Experiments
were tried with gauzes of different degrees of coarseness with
the same general result. The dischargingpower varies with
the coarseness of the gauze, and appears to depend more on
the ratio of the area of metal to the area of the openings than
on the actual size of the opening. If a copper gauze has such
a power of removing the carriers from the gas, we can readily
see why a small plug of cottonwool should completely abstract
the ions from the gas passing through it. The rapid loss of
conductivity is thus due to the smallness of the carrier and
the consequent rapidity of diffusion.
§ 19. General Remarks.
The cause and origin of the radiation continuously emitted
by uranium and its salts still remain a mystery. All the
results that have been obtained point to the conclusion that
uranium gives out types of radiation which, as regards their
effect on gases, are similar to Rontgen rays and the secondary
radiation emitted by metals when Rontgen rays fall upon
them. If there is no polarization or refraction the similarity
* Townsend, Phil. Mag. June 1898.
Notices respecting New Books. 163
is complete. J. J. Thomson * has suggested that the re
grouping of the constituents of the atom may give rise to
electrical effects such as are produced in the ionization of a
gas. Rontgen's j and Wiedemann's J results seem to show
that in the process of ionization a radiation is emitted which
has similar properties to easily absorbed Rontgen radiation.
The energy spent in producing uranium radiation is probably
extremely small, so that the radiation could continue for long
intervals of time without much diminution of internal energy
of the uranium. The effect of the temperature of the uranium
on the amount of radiation given out has been tried. An
arrangement similar to that described in § 11 was employed.
The radiation was completely absorbed in the gas. The
vessel was heated up to about 200° C; but not much differ
ence in the rate of discharge was observed. The results of
such experiments are very difficult to interpret, as the variation
of ionization with temperature is not known.
1 have been unable to observe the presence of any secondary
radiation produced when uranium radiation falls on a metal.
Such a radiation is probably produced, but its effects are too
small for measurement.
In conclusion, I desire to express my best thanks to
Prof. J . J. Thomson for his kindlv interest and encouragement
during the course of this investigation.
Cavendish Laboratory, Sept. 1st, 1898.
IX. Notices respecting New Books.
A Treatise on Dynamics of a Particle ; ivitli numerous examples. By
Dr. E. J. Eouth, F.R.S. Cambridge : University Press.
Pp. xii f 417.
npHAT this work is a thorough one on its subject is a matter of
* course, but it is more than this, it is a most interesting one. As
Dr. South remarks in the opening words of his preface, " so many
questions which necessarily excite our interest and curiosity are
discussed in the dynamics of a particle that this subject has always
been a favourite one with students." He puts the question, how
is it that by observing the motion of a pendulum we can tell the
time of rotation of the earth, or, knowing this, can deduce the
latitude of the place ? Other such problems excite our curiosity
at the very beginning of the subject. When we study the replies
to those problems we find new objects of interest, and so we
mount higher and higher until we include the planetary per
turbations, and take account of the finite size of bodies. So far
does Dr. Eouth carry us until he approximates quite closely to
his familiar Rigid Dynamics. One has hitherto associated his
work and name primarily with this latter subject, but as for
iome forty years the whole mathematical curriculum must have
jccupied his thoughts, he must have many potential books in his
* Proc. Camb. Phil. Soc. vol. ix. pt. viii. p. 397 (1898).
t Wied. Ann. lxiv. (1898). \ Zeit.f. Electrochemie, ii. p. 159 (1895).
164 Intelligence and Miscellaneous Articles.
mind, and we trust that his present leisure will enable him to
collect, into book form, the accumulated stores of these past
years. Our task is a simple one. "We shall merely indicate what
are the subjects discussed. There are in all eight chapters.
The headings are : — Elementary Considerations ; Rectilinear
Motion ; Motion of Projectiles ; Constrained Motion in Two
Dimensions ; Motion in Two Dimensions ; Central Forces ;
Motion in Three Dimensions ; and some Special Problems.
The work closes with two notes : the first on an Ellipsoidal
Swarm of Particles, and the second on Lagrange's Equations, a
new form for the Lagrangian function, and a rotating field. The
great value to the student appears to us to be the thorough
discussion of a large number of illustrative problems. As in his
previous books, Dr. Routh gives ample reference to original
memoirs, a number of historical notes, and a useful Index. Our
reference to the Preface sufficiently indicates the wide range
included under the heading Dynamics of a Particle.
X. Intelligence and Miscellaneous Articles.
SUSCEPTIBILITY OF D1AMAGNETIC AND WEAKLY MAGNETIC
SUBSTANCES.
To the Editors of the Philosophical Magazine.
Ge^tlbmeis,
TN the issue of the ' Philosophical Magazine'' for December 1898,
■ma note referring to the article published by me in the
May number 1898, " On the Susceptibility of Diamagnetic and
Weakly Magnetic Substances," Professor Quincke draws attention
to the fact that he had previously described* a method essentially
similar to the one I used for the determination of the susceptibility
and had applied it in investigating the susceptibility of Iron,
Nickel, and Cobalt, among other things.
That I failed to refer to this note of Professor Quincke's, and
to a communication by Lord Kelvin t on the same subject, was due
to the fact that I did not learn of them until after the article
referred to above had appeared.
I may be allowed to add that the method in the form used by
me, involving the use of prismatic slabs transversely magnetized,
does not admit of application to Iron, Nickel, and Cobalt, since in
this case the induced magnetization would depend almost altogether
upon the shape of the substance and but slightly upon its
susceptibility^. In Lord Kelvin's note upon this subject the
restriction that the method may be applied to those bodies only
which are diamagnetic or slightly magnetic is implied in the title.
Very truly yours,
Berlin, Dec. 10, 1898. Albert P. WlLLS.
* Tageblatt der 62 Vermmmlung der Deutsche?' Natur for seller und
Aerzte, Heidelberg, 1889, p. 209.
t " On a Method of determining in Absolute Measure, the Magnetic
Susceptibility of Diamagnetic and Feebly Magnetic Solids." — Report of
the British Association, 1890, p. 745.
\ Maxwell, ' Electricity and Magnetism,' vol. ii. pp. 65, 66.
PHL.Mag.S.5.Yol.47.Pl.I
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PHILOSOPHICAL MAGAZINE
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JOURNAL OF SCIENCE,
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FEBRUARY 1899. • : _,.^»
.*'' *
XL Radiation Phenomena in the Magnetic Field. — Magnetic
Perturbations of the Spectral Lines. By Thomas Preston,
M.A., D.Sc, F.R.S*
If
IN the April number of this Magazine f I described a
series of observations on u Radiation Phenomena in a *)j\
strong Magnetic Field. 5 '' Briefly stated, the results obtained
showed that while the majority of spectral lines became
triplets when the source of light was placed in the magnetic
field and viewed across the lines of force, yet this did not
hold good of all lines, for some were observed to be resolved
into quartets, or sextets, or other forms by the magnetic
field under precisely the same circumstances.
I pointed out at that time that these quartets &c.
might be regarded as modified forms of the normal or
standard triplet form, and might possibly be derived from it
by reversal. Thus, if each line of a triplet be reversed (that
is, if an absorptionband occurs along its middle), then we
have six lines instead of three, and so on for the other forms.
I also pointed out, however, that the general appearance of
these modified forms did not by any means favour that
explanation, for they possessed none of the ordinary cha
racteristics of reversals. Nevertheless, this explanation could
* Communicated by the Author.
t Phil. Mag. vol. xlv. p. 325 (1898). The experiments described in this
paper were performed in Oct. and Nov. 1897, and communicated to the
Royal Dublin Society in December 1897. See Trans. Rov. Dubl. Soc.
vol. xv. p. 385 (1898).
Phil Mag. S. 5. Vol. 47. No, 285. Feb. 1899. N
166 Dr. T. Preston on Radiation Phenomena
not be ignored till it had been proved by experiment that the
modifications were actually caused by other agencies, and
this I endeavoured to do by two lines of attack.
In the first place, if these phenomena are due to reversal, it
is likely that they will cease to exist when the quantity of
vapour in the source of light is greatly reduced. I accord
ingly tried sparking with weak solutions of salts instead of
with metallic electrodes, but in no case did the quartets, or
other modifications, reduce to the triplet form, but on the
contrary they became clearer and more precise as the lines
became sharper with the reduced quantity of vapour. Never
theless this was not regarded as conclusive or even seriously
in opposition to the supposition of reversal, for the appearance
of reversed lines in the strong magnetic field when the spark
is blown about might differ from that of an ordinary reversal.
I accordingly endeavoured to gradually increase the strength
of the magnetic field and observe if the components of the
supposed reversed line remained at the same distance apart
or became more widely separated as the strength of the field
increased. The extent to which I w 7 as able to increase the
field at that time was not, however, sufficient to enable me to
determine with sufficient certainty whether the reversal
hypothesis was tenable or not. For although the com
ponents of the supposed reversed line appeared to separate
under the increased field, yet this separation was not suffi
ciently great to overthrow the reversal hypothesis, for it
might be said that the absorbed band along the middle of the
line had merely become a little wider. The weight of
evidence, however, appeared to be against the reversal theory;
and in order to further test this matter I had a powerful
electromagnet specially built which it was hoped would
furnish a field sufficiently strong to determine matters
decisively, and in this respect it has not disappointed expect
ation. Thanks to the courtesy of the University Authorities
and of the Curator, Dr. W. E. Adeney *, I was able to resume
work at the Royal University with improved apparatus, and
it was soon found that the reversal theory must be abandoned
and that the explanation of the various deviations from the
normal triplettypes must be sought for in other agencies.
Before describing these more recent results, it will render
the explanation more intelligible if we refer for a moment to
figs. 1, 2, 3, 4, In fig. 1 the three lines A, B, C are
* I am deeply indebted to Dr. Adeney throughout this and the
previous investigations, for he invited me to the Royal University
laboratories and facilitated my work under conditions which necessarily
interfered with his own researches.
in the Magnetic Field.
167
supposed to represent the triplet into which a spectral line of
the standard type becomes resolved by the action of the
magnetic field, and fig. 2 in the same way represents a
quartet produced by the magnetic field, or, if we may say so,
A A
B A
B A C
Fi<?. 1.
Fiff. 2.
B
Fig. 3.
B C
Fig. 4.
a triplet in which the middle line A has become a doublet.
These lines are all planepolarized, the vibrations in the side
lines B and C being parallel to their length, while the
vibrations in the central constituents A are in the per
pendicular direction, when the light is viewed across the lines
of force.
Hence, if a doubleimage prism be placed before the slit of
the spectroscope in the path of the beam of light, the two
planepolarized parts can be separated so that one part (say
A) forms one image on the slit, while the other part (B and
C) forms another image on the slit. These two images, being
separated, give rise to two spectra in the field of view of the
spectroscope which may be separated or may be caused to
partially overlap if so desired. As a consequence a triplet or
a quartet which appears in the field of view, as shown in
figs. 1 and 2, without the doubleimage prism, becomes trans
formed by the doubleimage prism into figs. 3 and 4. Thus
the light vibrating parallel to the lines of force (A) is
separated from that vibrating in the perpendicular direction
(B and C), and this facilitates observation in the case of small
separations and in the case of overlapping lines. This sepa
ration of the two parts can of course be effected by a Nicol's
prism, but the doubleimage prism has the advantage of
showing the two parts simultaneously.
The question now before us is — can the quartet shown in
figs. 2 and 4 be derived from the triplets figs. 1 and 3 by
mere reversal of the central line A. In answer to this
question, it is to be remarked that it is not the central line
A alone which shows as a doublet, for in some cases the
N2
168 Dr. T. Preston on Radiation Phenomena
side lines B and C show as doublets and in others as triplets:
and again, in some quartets the distance between the central
pair A is almost as great as, and it may be greater than, the
distance between the side lines B and C. Further, as the
magnetic field increases in strength, the distance between the
members of the central pair A (fig. 4) increases at the same
rate as the distance between the side lines B and C. Again,
when the side lines B and C are each resolved into doublets
(or triplets) the separation of the constituent lines of each of
these doublets (or triplets) increases with the magnetic field
like the separation of the components of the normal triplet
ABC shown in fig. 1 ; and in face of these facts the reversal
theory becomes quite untenable.
The general phenomenon, therefore, which remains to be
explained is the further resolution of each constituent of the
normal triplet into a doublet or a triplet or some other system;
and, as we shall see immediately, the electromagnetic theory
proposed by Dr. Larmor * may be extended to embrace all
the phenomena yet observed. Before proceeding to consider
this explanation, however, it is necessary to refer to a
particular case which was recently announced as having been
observed by MM. Becquerel and Deslandresf, and subse
quently by Messrs. J. S. Ames, R. F. Earhart, and H. M.
Reese J, and which they refer to as an example of " reversed
polarization." This phenomenon is represented to be as
follows.
Consider the triplet shown in fig. 1 ; then in the normal
state of affairs the vibrations in A are parallel to the lines of
force, w r hile the vibrations in B and C are perpendicular to
the lines of force. Now in the spectrum of iron the authors
just named have recorded that they observed triplets in which
the vibrations in the middle line were perpendicular to the lines
of force, while the vibrations in the side lines were parallel to
the lines of force — the reverse of the normal case. Stated in
this way the phenomenon is very startling, and appears at
first sight to be directly contradictory to all theoretical
expectation. But if we return to fig. 4, it will be seen at
once that this phenomenon, supposing it to exist, can be
regarded merely as an extreme case of the quartet. For, as
we have already said, the horizontally vibrating lines A of
the quartet may be close together or widely separated. They
may be even more widely separated than the vertically
vibrating fines B and C (fig. 5), and in some particular cases
* Dr. J. Larmor, Phil. Mag;, vol. xliv. p. 505 (1897).
t Comptes Rendus, t. cxxvi. p. 997, April 4th, 1898.
% AstroPhysical Journal, vol. viii. p. 48, June 1898.
in the Magnetic Field. 169
B and C may be very close together, or coincide, while the
centre pair A are separated by a considerable space, as shown
in fig. 6. In this extreme case we are furnished with a
triplet in which the centre as it were encloses the sides.
But this is no specially new form, being quite continuous
with the other types of modification.
Fig. 5.
A
Fig. 6.
A
B
BC
Once the doubling of the centre line (A, figs. 2 or 4) is
explained, the other types follow in sequence as expected
variations, for the cause which converts A into a doublet
may be sufficiently powerful to separate the constituents of A
more widely than B is separated from C, and the separation
of the constituents of A might be tolerably large, even
though the separation of B and C is quite insensible.
Thus, if the socalled reversed polarization is shown by any
lines, the explanation offers no difficulty once we have ex
plained the quartet, but it is doubtful if the lines indicated in
the spectrum of iron by the French and American observers
just mentioned show this peculiarity. Iron was one of the
first substances which I examined *, because I considered . it
might present peculiarities, but I did not observe in it any
marked differences from the behaviour of other substances.
Several quartets and other slight modifications occur, but the
lines referred to by the French and American observers do
not, on my photographic plates, exhibit the exact peculiarity
attributed to them. The central part corresponding to A in
fig. 4 is a doublet without doubt, but the remainder (corre
sponding to the lines B and C, fig. 4) does not appear to be by
any means a single line, but looks rather like a triplet of which
* See Proc. Koyal Society, January 1898. The doublets referred to in
this paper turned out on analysis to be quartets.
170 Dr. T. Prestori on Radiation Phenomena
the side lines are broad and weak, while the centre is much
denser. It is just possible, and indeed probable, that these
modifications may be really quartets, of which the side lines
B and C are broad and weak, and overlap at their inner
edges, giving the appearance of a bright central line winged
with two weaker bands.
The distance between the side lines B and is about
the same as that between the components of A, and when the
doubleimage prism is not used, the lines in question photo
graph as triplets, i. e. as bands having three ribs or denser
parts running along them lengthwise. With a much stronger
field it could be determined whether the part BC in these
lines is really a triplet or an overlapping doublet; but as they
are all weak lines requiring long exposure (four hours in my
case), it is not easy to arrange to have a very strong field for
such a long time. However, it is a matter of very little
importance at present, for if we can explain the quartet we
are on the highway to the explanation of all the various
modifications.
For this purpose let us revert to Dr. Larmor's paper
already cited. In this investigation he considers merely the
simple case of a single ion describing an elliptic orbit under
a central force directly proportional to the distance. This
electric charge, when subject to the influence of the magnetic
field, is so acted on that its elliptic orbit is forced into
precession round the direction of the magnetic force, that is,
as a first approximation. For the equations of motion of the
ion moving round a centre of force in a magnetic field are,
as a first approximation, the same as those which obtain for
a particle describing an elliptic orbit under a central force
when the orbit precesses or revolves round a line through its
centre drawn in the direction of the lines of magnetic force.
If N be the natural frequency of revolution of the particle in
its orbit and n the frequency of revolution of the orbit in its
precessional movement, the combined movement is equivalent
to three coexisting motions of frequencies N + w, N, and
'N — n respectively. When n is small compared with N so
that its square may be neglected, the equations of motion of
the particle in the revolving orbit become identical with those
of the moving ion in the magnetic field.
This simple theory therefore predicts that a single spectral
line should be converted into a triplet by the action of the
magnetic field, and that the constituents of this triplet should
be planepolarized when viewed across the lines of force. It
teaches us that the cause of the tripling is the forced precession
of the ionic orbits round the lines of magnetic force, and it
in the Magnetic Field, 171
assigns a dynamical cause for this precession in the action of
the magnetic field on the ionic charge moving through it.
But up to this point the electromagnetic solution deals
with a perturbation which is really not the full equivalent
of a precessional movement of the orbit, and therefore the
investigation given by Dr. Larmor applies, as he himself
states, to a single simple case. For the equations of motion
of a particle describing under a central force an elliptic orbit
which precesses with angular velocity co round a line whose
directioncosines are (/, m, n) are
x=—Q / 2 x + 2(o(ny — mz) + co^x — (o' 2 l(l% + my + nz), . (1)
with two similar equations for y and z, whereas the equations
of motion of the ionic charge moving under a central force in
a magnetic field are, as given by Dr. Larmor,
x=—£l 2 %\k(ny—mz), .... (2)
with two similar expressions for y and z. The latter equation
coincides with the former if we neglect co 2 and take 2a>=#,
that is, when the precessional motion is relatively small.
The motion imposed on the ion by the electromagnetic
theory is therefore merely a simple type of precessional pertur
bation of the orbit, and, as other perturbations may occur,
and indeed ought to be expected to occur, it is clear that the
simple triplet is not the only form which we should expect to
meet with when the matter is investigated experimentally.
Thus, if the orbit besides having a precessional motion has in
addition an apsidal motion, that is a motion of revolution in
its own plane, then each member of the triplet arising from
precession will be doubled, and we are presented with a
sextet as in the case of the D 2 line of sodium. Similarly, if
the inclination of the plane of the orbit to the line round
which precession takes place be subject to periodic variations,
then each member of the precessional triplet will itself become
a triplet, and so on for other types of perturbation.
It is quite unnecessary to enter into these matters in any
detail here, for the whole explanation was fully given and
published in 1891 by Dr. G. J. Stoney *, that is, six years
before the effects requiring explanation had been observed.
Dr. Stoney's aim was to explain the occurrence of doublets
and equidistant satellites in the spectra of gases, that is in
the normal spectra unaffected by the magnetic field — for at
* This most important paper of Dr. Stoney's was published in the
Scientific Transactions of the Royal Dublin Societ}^ vol. iv. p. 563
(1891), " On the Cause of Double Lines and of Equidistant Satellites in
the Spectra of Gases."
172 Dr. T. Preston on Radiation Phenomena
that time the influence of the magnetic field was not known
to exist. The character of certain spectra indicated that the
lines resolved themselves naturally into groups, or series. For
example, in the monad elements Na, K, &c. the spectrum
resolves itself into three series of doublets like the D doublet
in sodium, and Dr. Stoney's object was to explain the exist
ence of these pairs of lines. For this purpose he considered
what the effect would be on the period of the radiations from
a moving electron if subject to disturbing forces. In the
first place he determined that if the disturbing forces cause
the orbit to revolve in its own plane, that is, cause an apsidal
motion, then each spectral line will become a doublet. The
frequencies of the new lines will be N + ft and N — n, where N
is the frequency of the original line and n the frequency of
the apsidal revolution. This is very easily deduced by
Dr. Stoney from the expressions for the coordinates of the
moving point at any time t. Thus if a particle describes an
ellipse under a force directed towards its centre (law of direct
distance), its coordinates at any instant are
x = a cos fit, y — b sinl2£,
in which fl is equal to 27rISr, where N is the frequency of
revolution. But if, in addition, the ellipse revolves around
its centre in its own plane with an angular velocity co, it is
easily seen by projection that the coordinates at any time
are
x = a cos fit cos cot — b sin fit sin cot,
y = a cos H t sin cot + b sin fit cos cot ,
and these are equivalent to
# = i(a + h) cos (fl + co)t + i(a — b) cos (12 — a>)t,
y — ^{a[b) sin (fl + co)t — i(a — 6) sin (fl — co)t,
and these in turn are equivalent to the two opposite circular
vibrations
x x = J(a + b) cos (12 + ©)£ "1 x 2 = i(a — b) cos (12  »)n
j/i = i(a + &) sin (fl + co)t J y 2 = — J(a— 6) sin (12 — co)t J '
The resultant motion is consequently equivalent to two
circular motions in opposite senses of frequencies N + n and
N — n.
This is an analysis of the motion without any regard to the
dynamical origin of it ; but if we treat it from a dynamical
point of view, the equations of motion will exhibit the forces
which are necessary to bring about the supposed motion.
Thus, if the orbit rotates with angular velocity o> in its own
in the Magnetic Field. 173
plane while the particle is attracted to a fixed centre with a
force OV 3 then, by taking the moving axes of the orbit as
axes of reference, the equations of motion are
x — — £l 2 x + co 2 x + 2coy. \ , „.
y = — 12^+ coy— 2 cox, J
so that if (#, y)=e ipt be a solution, we have at once
p = £l ±co,
which shows the doubly periodic character of the motion, and
also exhibits the character of the perturbing forces necessary
to produce the given apsidal motion of the orbit. For if the
orbit were fixed, the equations of motion would be (x } y)
= —£l 2 (x,y): hence the remaining terms on the righthand
side of the above equations must represent the perturbing
forces. Of these the final terms 2 coy and —2 cox are the x
and y components of a force 2cov, where v is the velocity of
the particle, acting in a direction perpendicular to v 9 that is
along the normal to the path of the particle, and represent
the forces which a charged ion would experience in moving
through a magnetic field with the lines of force at right
angles to the plane of the orbit, if 2co be taken equal to k in
Dr. Larmor's equations (2). The other pair of terms, co 2 x and
co 2 y, represent a centrifugal force arising from the imposed
rotation co. If we neglect g> 2 , the above equations become
identical with those which hold in Larmor's theory for the
moving ion, as they obviously should, for an apsidal motion in
the plane of the orbit is the same thing as a precession about a
line perpendicular to the plane of the orbit, and in this case
there will be no component in the direction of the axis round
which precession takes place ; accordingly the middle line of
the precession triplet will be absent, and we are furnished
merely with a doublet.
Now in the magnetic field the perturbing force, being the
magnetic force, is fixed in direction, and on this account the
doublets and triplets arising from perturbations caused by it
are polarized. On the other hand, if the perturbing forces be
not constant in direction, this polarization should cease to
exist, and polarization should not be expected in the case of
any lines of the normal spectrum, even though these happen
to be derived from other lines by perturbations in the manner
conceived by Dr. Stoney.
In the same way the general case of precessional motion
may be worked backwards in order to discover the types of
force which produce the perturbation. Thus, taking the axes
174 Dr. T. Preston on Radiation Phenomena
moving with angular velocity co round a line whose direction
cosines are /, m, n, the equations of motion
x = — fl 2 x + Zco(ny — mz) + co 2 x — co 2 l(lx\ my + nz) ,
" = — Q?y + 2co(lz — nx) + co 2 y — co 2 m(lx J my + nz), J> (4)
z = — O 2 ? + 2co(mx — ly) + co 2 z — co' 2 n(lx + my f nz)
show that the perturbing forces consist firstly of a force
2cov sin 6 ; where v is the velocity of the particle and 6 the
angle which its direction makes with the axis round which the
precession co takes place. This force acts along the normal to
the plane of v and co (the direction of the velocity and the axis
of rotation), and is precisely the force experienced by an
ion moving in a magnetic field in Larmor's theory. The
remaining terms containing co 2 are the components of a
centrifugal force arising from the rotation round the axis
(I, m, n), and this is negligible only when co is relatively
small.
If the direction I, m, n be taken to be that of the lines of
magnetic force, and if the axis of z be taken to coincide with
this direction, then the equations (4) simplify into
x=(Q?co' ;L )x + 2coy~\
y=(Wco*)ij2cox\ (5)
s = n% j
and these are the equations of motion of a particle describing
an elliptic orbit which precesses with angular velocity co round
the axis of z. The two first of these equations contain x and
y and give the projection of the orbit on the plane x, y, at
right angles to the axis of the magnetic field. This pro
jection is an ellipse revolving in its own plane with an apsidal
angular velocity co, and gives rise to the two side lines of the
normal triplet of frequencies (£1 ± co)I'Itt. On the other hand,
the vibration parallel to the axis of z is unaffected by the
precessional motion and gives rise to the central line of the
triplet of frequency H/2ir.
Now in order to account for the quartet (fig. 2) we must
introduce some action which will double the central line A
while the side lines B and C are left undisturbed. That is,
we must introduce a double period into the last of equations
(5) while the first and second remain unchanged. This is
easily done if we write the equation for z in the form
2 = AsinX2£ . . . . . ... (6)
in the Magnetic Field. 175
and remark that this will represent two superposed vibrations
of different periods, if we regard A as a periodic function of
the time instead of a constant. That is, if we take A to be
of the form a sin nt, we shall have
z = a sin nt sin Clt = p[GOS (ft — n)t — cos {Q, + n)t] }
which represents two vibrations of equal amplitude and of
frequencies (ft — ?i)/2tt and (ft + w)/27r as required to produce
the quartet. The magnitude of n determines whether the
separation of the constituents of the central line A (fig. 2)
shall be less than, or greater than, the separation of the side
lines B and C, and if the former is sensible while the latter is
insensible we are presented with the case depicted in fig. 6 —
although, as I have said before, my observations do not
confirm the existence of this case.
The supposition made above to account for the doubling of
the middle line, viz. that the amplitude of the z component of
the vibration varies periodically, is one which appears to be
justified when we consider the nature of the moving system and
the forces which control it. For the revolving ion is part of
some more or less complex system which must set in some
definite way under the action of the magnetic field — say with
its axis along the direction of the magnetic force — and, in
coming into this position, the inertia of the system will cause it
to vibrate with small oscillations about that position of equi
librium, and this vibration superposed on the precessional
motion of the ionic orbit gives the motion postulated above
to explain the quartet.
This, indeed, comes to the same thing as a suggestion made
by Professor Gr. F. FitzGrerald about a year ago — shortly
after I discovered the existence of the quartet form (Oct. 1897).
In Professor Fitz Gerald's view, the ion revolving in its orbit is
equivalent to an electric current round the orbit, and therefore
the revolving ion and the matter with which it is associated
behave as a little magnet having its axis perpendicular to the
plane of the orbit. The action of the magnetic field will be to
set the axis of this magnet along the lines of force, and in
taking up this set the ionic orbit will vibrate about its position
of equilibrium just as an ordinary magnet vibrates about its
position of rest under the earth's magnetic force.
In a similar way a periodic change in the eliipticity of the
orbit produces a doubling of the lines, while a periodic
oscillation in the apsidal motion renders the line nebulous or
diffuse; and by treating these cases in the foregoing manner
■the corresponding forces may be discovered. It is clear,
176 Dr. T. Preston on Radiation Phenomena
therefore, that perturbations of this kind are sufficient to
account for all the observed phenomena, and, further, that
perturbations of this kind are almost certain to be in
operation throughout some, at least, of the ionic motions.
The existence of all these variations of the normal triplet
type are therefore of great interest, not only in showing that
the perfect uniformity required for the production of the
normal triplet is not maintained, as we should expect, in all
cases, but also as an experimental demonstration that the
causes supposed by Dr. Stoney, in 1891, to be operative in
producing doublets and satellites in the natural spectra of
gases may be really the true causes by which they are produced.
Nevertheless Dr. Stoney's explanation of the natural doublets
is opposed by a serious difficulty in the fact that the two lines
of a given doublet, say the two D lines of sodium, behave in
different ways, as if they arose from different sources rather
than from the perturbation of the same source. For, in
addition to the differences previously known to exist, there is
the difference of behaviour in the magnetic field. Thus T) x
is a widemiddled quartet in which the distance between the
central lines A (fig. 4) is nearly as great as the distance
between the side lines B and C, while D 2 shows as a sextet of
uniformly spaced lines.
In a similar manner individual members of the natural
triplets which occur in the natural spectra of the zinc,
cadmium, magnesium, &c. group behave differently. Thus
if we denote the members of one of the natural triplets
by the symbols T 1? T 2 , T 3 , in ascending order of re Iran 
gibility (for example the triplet 5086, 4800, 4678 of
cadmium, or the triplet 4811, 4722, 4680 of zinc, or the
green b triplet of magnesium), we find that T 3 in all cases, in
the magnetic field, shows as a pure triplet, or suffers accord
ing to the foregoing merely precessional perturbation. On
the other hand, T 2 shows in each case as a quartet while T 1 is
a diffuse triplet in which each of the members may prove to
be complex on further resolution. This would seem to point
to an essential difference in the characters of the lines T 1? T 2 , T 3 ,
as if they sprang from different origins rather than immedi
ately from the same. It is also of great interest to note that,
so far as my observations yet show, these natural triplets
behave differently according as they belong to Kayser and
Runge's first subsidiary series or to the second subsidiary
series. Thus if the triplet T 1? T 2 , T 3 , belongs to the first
subsidiary series, then the magnetic effect decreases from T 2
to T 3 , while if it belongs to the second subsidiary series, the
magnetic effect increases from T x to T 3 . Examples of this
in the Magnetic Field. 177
latter class are shown in my communication to this Journal
(April 1898, p. 335), where the increasing character of the
magnetic effect is well exhibited in the natural triplets 5086,
4800, 4678 of cadmium, and 4811, 4722, 4680 of zinc.
Further examples of this, and other peculiarities, I hope to
give in the near future as soon as I have fully examined and
verified them.
General Law.
The first general survey of the magnetic effect on the
spectral lines of any given substance did not appear to favour
the view that the phenomena are subject to any simple law.
According to the electromagnetic theory the separation, S\,
of the side lines of a magnetic triplet should, under the same
conditions, vary directly as X 2 as we pass from line to line of
the same spectrum. The possibility of such a law as this
seemed to be refuted by the fact that some lines are largely
affected in the magnetic field while others, of nearly the same
wavelength in the same spectrum, are not appreciably affected
under the same circumstances. In this connexion, however,
I pointed out * that " it is possible that the lines of any one
substance may be thrown into groups for each of which hX
varies as A. 2 , and each of these groups might be produced by
the motion of a single ion. The number of such groups in a
given spectrum would then determine the number of different
kinds of ions in the atom or molecule.
" Homologous relations may also exist between the groups
of different spectra, but all this remains for complete in
vestigation."
Although the investigation referred to in the foregoing is
still far from complete, yet the measurements so far made
uniformly tend to confirm the above speculation. For the
corresponding lines of the natural groups into which a given
spectrum resolves itself possess the same value of e/m or 8X/X 2 3
and, further, this value is the same for corresponding lines in
homologous spectra of different substances.
To illustrate the meaning of this, take the case of mag
nesium, cadmium, and zinc, which are substances possessing
homologous spectra and belonging to the same chemical
group (MendelejefFs second group). The spectra of these
metals consist of series of natural triplets. The first triplet
of the series in magnesium is the green b group consisting
of the wavelengths 5183*8, 51728, 5167'5 ; while the first
cadmium triplet consists of the lines 5086, 4800, 4678, and
Phil. Mag., April 1898, p. 337.
1 78 On Radiation Phenomena in the Magnetic Field.
the first zinc triplet consists of the lines 4810*7, 4722, 4680.
Each of these triplets belongs to Kayser and RungVs second
Nebenserie, being the first terms, corresponding to n = 3, in
their formula. We should consequently expect these groups
to behave similarly in the magnetic field and to show effects
which are similar for corresponding lines. That this expect
ation is realized is shown by the following table : —
Magnesium.
Cadmium.
Zinc. mje or \ 2 /8\.
Character.
51838
5086
48107 18 approx.
Diffuse triplets.
51728
4800
4722 115 „
Quartets.
51675
4678
4680 10
Pure triplets.
Thus the corresponding lines 5183'8, 5086, and 4810*7 of
the different substances possess the same value for m/e, while
the other corresponding lines also possess a common value
for the quantity m/e. The value of this quantity changes
from one set of lines to another, showing, as we should expect,
that the different sets arise from differences in the source
which produces them.
Not only is the quantity m/e the same for corresponding
lines in homologous spectra, but, as shown in the above table,
the character of the magnetic effect is also the same for
corresponding lines. Thus, while the lines along the lowest
row, 51675, 4678, 4680, are all of the pure triplet type, the
lines of the middle row all become resolved into similar
quartets in the magnetic field, and the lines forming the top
row are all somewhat diffuse and show as " soft " triplets of
which the constituents may be really complex on further
resolution.
It thus appears that the observation of radiation pheno
mena in the magnetic field is likely to afford a valuable means
of inquiry into the so far hidden nature of the events which
bring about the radiation from a luminous body, and also to
give us, perhaps, some clearer insight into the structure of
matter itself.
[ 179 ]
XII. On the Reflexion and Refraction of Solitary Plane
Waves at a Plane Interface behveen two Isotropic Elastic
Mediums — Fluid, Solid, or Ether . By Lord Kelvin,
G.C.V.O*
§ 1. "rC'LASTIC SOLID" includes fluid and ether; except
J__i conceivable dynamics f of the mutual action
across the interface of the two mediums. Maxwell's electro
magnetic equations for a homogeneous nonconductor of
electricity are identical with the equations of motion of an
incompressible elastic solid J, or with the equations expressing
the rotational components of the motion of an elastic solid
compressible or incompressible ; but not so their application
to a heterogeneous nonconductor or to the interface between
two homogeneous nonconductors §.
§ 2. The equations of equilibrium of a homogeneous elastic
solid, under the influence of forces X, Y, Z, per unit volume,
acting at any point (x, y, z) of the substance are given in
Stokes' classical paper u On the Theories of the Internal
Friction of Fluids in Motion, and of the Equilibrium and
Motion of Elastic Solids/' p. 115, vol. i. of his ' Mathematical
Papers': also in Thomson and Tait's ' Natural Philosophy'
[§698(5) (6)]. Substituting according to D'Alembert's
principle, — pf, — py, —pi for X, Y, Z, and using as in
a paper of mine ] of date Nov. 28, 1846, V 2 to denote
d 2 d l d 2
the Laplacian operator 73 + j— 2 + 'J^ii we ^ n< ^ as the equa
tions of motion x & z
pS =(  k+ ° n)d £ +nv ^
* Communicated by the Author j having been read before the Royal
Society of Edinburgh on December 19, 1898.
t See Math, and Phys. Papers, vol. iii. art. xcrx. (first published
May 1890), §§ 1420, 2128; and particularly §§ 4447. Also Art. c.
of same volume ; from Comptes Rendus for Sept. 16, 1889, and Proc
Roy. Soc. Edinb., March 1890.
\ See ' Electricity and Magnetism,' last four lines of § 616, last four
lines of § 783, and equations (9) of § 784.
§ Ibid. §611, equations (1*). In these put 0=0, and take in con
nexion with them equations (2) and (4) of § 616. Consider K and u as
different functions of x, y, z ; consider particularly uniform values for
each of these quantities on one side of an interface, 'and different uniform
values on the other side of an interface between two different non
conductors, each homogeneous.
 Camb. and Dublin Math. Journal, vol. ii. (1847). Republished as
Art. xxvn., vol. i. of Math, and Phys. Papers.
180 Lord Kelvin on the Reflexion
p denoting the density of the medium, f , rj, f its displacement
from the position of equilibrium (#,y,s), and 8 the dilatation
of bulk at (a, y, z) as expressed by the equation
*2?J+2 0*
§ 3. Taking d/dx, d/dy, d/dz of (1), we find
d' 2 S
pjr== (*+*»)?* • (3).
From this we find
v * 8 = Ltfc.(Jp (4 ,
Put now
e«H£v*i • * + £ V 2 S; l6 + £^I (5).
These give
fi + p + f = (6)
and therefore, eliminating by them £. 9;, £ from (1), we find
by aid of (4),
p d S= n ^> p c w= n ^> p d S= n ^> w
§ 4. By Poisson's theorem in the elementary mathematics
of force varying inversely as the square of the distance,
we have
V~ 2 S = ^; J J J d (volume) • pp ; (8) ,
where S, 8' denote the dilatations at auy two points P and P ; ;
d (volume) denotes an infinitesimal element of volume around
the point P x ; and PP' denotes the distance between the points
P and P'. This theorem gives explicitly and determinately
the value of V _2 8 for every point of space when o is known
(or has any arbitrarily given value) for every point of space.
§ 5. If now we put
f2 =jv 2 s; % =v 2 a ; &=va ; (9) ,
we see by (5) that the complete solution of (1) is the sum of
two solutions, (£ 1? %, fi) satisfying (6) and therefore purely
distortional without condensation; and (f 2 , %, f 2 ), ivhich, in
and Refraction of Solitary Plane Waves. 181
virtue of (9) , is irrotational and involves essentially rarefaction
or condensation or both. This most important and interesting
theorem is, I believe, originally due to Stokes. It certainly
was given for the first time explicitly and clearly in § § 58
of his " Dynamical Theory of Diffraction '"'*.
§ 6. The complete solution of (3) for plane waves travelling
in either or both directions with fronts specified by (a, ft 7),
the directioncosines of the normal, is, with ty and % to denote
arbitrary functions,
s = ^(,_^±to) +x (, + f£±to) (10) ,
where
e=?v /*+t? (11) .
so that v denotes the propagationalvelocity of the con
den sationalrarefactional waves. By inspection without the
aid of (8), we see that for this solution
v^K!H*H**^ + x(. + =**±*!)i]
For our present purpose we shall consider only waves
travelling in one direction, and therefore take %=0; and,
(d \ — *
T.J f
instead of v( rj ^ ;f being an arbitrary function. Thus
by (12) and (9) we have, for our condensationalrarefactional
solution,
k_ 7 h__k_, f( t gg±gy±3? \ . . (13).
* ft y J \ V J
In the wavesystem thus expressed the motion of each
particle of the medium is perpendicular to the wave front
(a, ft 7). For purely distortional motion, and wavefront
still (a, ft 7) and therefore motion of the medium everywhere
perpendicular to (a, ft 7), or in the wavefront, we find
similarly from (7) and (6)
fi  vi _ Sl rls ax+ fy+v z \ (u),
^A" i 8B7C" / V u / '
where
••■V7 (15) '
* Camb. Phil. Trans., Nov. 26, 1849. Republished in vol. ii. of his
' Mathematical Papers.'
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899.
182 Lord Kelvin on the Reflexion
and so denotes the propagational velocity of the distortional
waves ; and A, B, C are arbitrary constants subject to the
relation
«A + /3B + 7C = (16).
§ 7. To suit the case of solitary waves we shall suppose the
arbitrary function f(t) to have any arbitrarily given value for
all values of t from to t, and to be zero for all negative
values of t and all positive values greater than t. Thus t is
what we may call the transittime of the wave, that is, the
time it takes to pass any fixed plane parallel to its front ; or
the time during which any point of the medium is moved by
it. The thicknesses, or, as we shall sometimes say, the wave*
lengths, of the two kinds of waves are ur and vr respectively,
being for the same transittimes directly as the propagational
velocities.
§ 8. And now for cur problem of reflexion and refraction.
At present we need not occupy ourselves with the case of
purely distortional waves with vibratory motions perpendicular
to the plane of the incident, reflected, and refracted rays. It
was fully solved by Green * with an arbitrary function to
express the character of the motion (including therefore the
case of a solitary wave or of an infinite procession of simple
harmonic waves). He showed that it gave precisely the
" sine law " which Fresnel had found for the reflexion and
refraction of waves " polarized in the plane of incidence."
The same law has been found for light, regarded as electro
magnetic waves of one of the two orthogonal polarizations,
by von Helmholtz, H. A. Lorenz, J. J. Thomson, Fitz Gerald,
and Rayleigh f . None of them has quite dared to say that
the physical action represented by his formulas for this case is
a toandfro motion of the ether perpendicular to the plane of
incidence, reflexion, and refraction ; nor has any one, so far
as I know, absolutely determined whether it is the lines of
electric force or of magnetic force that are perpendicular
to that plane in the case of light polarized by reflexion at the
surface of a transparent medium. For the action, whatever
its rjhysical character may be, which takes place perpendicular
to that plane, they all seem to prefer " electric displacement/'
of which the only conceivable meaning is motion of electricity
to and fro perpendicular to the plane. If they had declared,
or even suggested, definitely this motion of ether, they would
* " On the Reflexion and Refraction of Light at the common Surface
of two NonCrystallized Media/' Math. Papers, p. 258. Also Trans.
Caruo. Phil. Soc. 1838.
t Sde Glazebrook's Rsport " on Optical Theories " to British Asso
ciation, 1885.
and Refraction of Solitary Plane Waves.
183
have been perfectly in harmony with the undulatory theory of
light as we have it from Young and Fresnel. We shall return
to this very simple problem of reflexion and refraction of
purely distortional waves in which the motion is perpendicular
to the plane of the three rays, in order to interpret in the very
simplest case the meaning, for a solitary wave, of the " change
of phase " discovered by Fresnel and investigated dynamically
by Green for a procession of periodic waves of simple harmonic
motion experiencing " total internal reflexion/' (See § 20
below.)
§ 9. Meantime we take up the problem of the four reflected
and refracted waves produced by a single incident wave of
purely distortional character, in which the motion is in a
plane perpendicular to the five wavefronts. Taking this for
XOY, the plane of our diagram, let YOZ be the interface
between the two mediums. We shall first consider one
single incident wave, I, of the purely distortional character.
By incidence on the interface it will generally introduce
reflected and refracted waves 1', I y , of its own kind, that is
purely distortional, and J', J r reflected and refracted waves
Fm. 1,
of the condensationalrarefactional kind. The diagrams re
present, for two cases, sections of portions of the five waves
by the plane XOY. F and R show the front and rear of
each wave ; and the lines of shading belonging to it show
the direction of the motion, or of the component, which it
02
184
Lord Kelvin on the Reflexion
gives to the medium. The inclinations of the fronts and rears
to OX, being what are ordinarily called the angle of incidence
FiK. 2.
and the angles of reflexion or refraction of the several waves,
will be denoted by i, i', i n j,j r The value of 7 for each of
the five waves is zero, and the values of a and ft are as shown
in the following table :—
a.
S.
I
j sin i
— cos i
r
+ sin i
+ cos i
— cos i,
1
+ sin i t
L
+ siiy"
+ cos^'
j
+ sinj?;
 cosj>*,
The section of the five waves by OX is the same for all,
being expressed by ur/sin i for I, and by corresponding
formulas for the four others. Hence if we denote rtimes its
and Refraction of Solitary Plane Waves. 185
reciprocal by a, we have
a _ sin i _ sin /, __ shy' _ sin,/,
where u and ?/, are the propagational velocities of the distor
tional waves, and v, v n those of the condensational waves in
the two mediums. If now we take
b = acoti= \Z(ii 2 —a 2 ) ; b l = acoti l = s/(u~ 2 — a 2 ) ;
c=acotj= \Z(v~ 2 — a 2 ) ; e, = acotj t = \/(v t ~ 2 — « 2 ); (18),
we have for the arguments of/ in the five waves
t— ax + by; t—ax — by; t — ax + b t y; t — ax — cy; t — ax + Cjy (19),
§ 10. Following Green * in calling the two sides of the
interface the upper and lower medium respectively (and so
shown in the diagram), we have for the components of the
displacement in the upper medium
g=blf(t — ax + hy)—blf (t — ax — by) + aJ'f (t — ax — cy) 
r) = alf(t — ax + by)+al'f(t — axby) +cJ'f[t — axcy) J ^' ''
and in the lower medium
^bLfitax+b^raJJitax + cy) 
rj = al,f(t — ax + b,y) — c t JJ(t — ax+ c,y) )
where I, I', I,, J', J p denote five constant coefficients. The
notation J' and J, is adopted for convenience, to reserve the
coefficient J for the case in which the incident wave is con
densational, and there is no incident distortional wave.
There would be no interest in treating simultaneously the
results of two incident waves, one distortional (I) and the
other condensational (J).
§ 11. We may make various suppositions as to the inter
facial conditions, in respect to displacements of the two
mediums and in respect to mutual forces between them.
Thus we might suppose free slipping between the two : that
is to say, zero tangential force on each medium; and along
with this we might suppose equal normal components of
motion and of force ; and whatever supposition we make as
to displacements, we may suppose the normal and tangential
forces on either at the interface to be those calculated from
the strains according to the ordinary elastic solid theory, or
to be those calculated from the rotations and condensations or
dilatations, according to the ideal dynamics of ether suggested
in the article referred to in the first footnote to § 1. We shall
* Green's ' Math. Papers/ p. 253.
(22).
186 Lord Kelvin on the Reflexion
for trie present take the case of no interfacial slip, that is, equal
values of g, 7] on the two sides of the interface. Remarking
now that the argument of /for every one of the five waves is
t — ax where y = 0, we see that the condition of equality of
displacement on the two sides of the interface gives the
following equations : — ■
6(11') +oJ / =& / I / + aJ i TJ
a (I + 1') + c J' = al J — c l J l f
§ 12. As to the forceconditions at the interface, I have
already given, for ordinary elastic solid or fluid matter * on
the two sides of the interface, a complete solution of the
present problem in my paper f "On the Eeflexion and
Refraction of Light " in the ' Philosophical Magazine ' for
1888 (vol. xxvi.) ; nominally for the case of simple harmonic
wavemotion, but virtually including solitary waves as
expressed by an arbitrary function: and I need not now
repeat the work. At present let us suppose the surfaceforce
on each solid to be that which I have found it must be
for ether J, if magnetic force is due to rotational displace
ment of ether, and the lines of magnetic force coincide with
axes of rotation of etherial 'substance. According to this
supposition the two components, Q (normal) and T (tan
gential), of the mutual force between the mediums, which
must be equal on the two sides of the interface, are
Q=«(f + ?)
\dx dy) ™
\dx dy) '" J \dx dy.
where k denotes for ether that which for the elastic solid we
have denoted by (k +■ fft), and suffixes indicate values for the
lower medium. If we begin afresh for ether, we may define
n as l/4«r of the torque required to hold unit of volume
of ether rotated through an infinitesimal angle us from its
* The forceconditions for this case are as follows : —
Normal cooiponent force equated for upper and lower mediums,
C*f.)«+a.5=(t •»>,+*., (J) /S
and taDgential forces equated,
•ffi+fHS+D.
t In that paper B, A, and £ denote respectively the n, the k\pi, and
the p of the present paper.
J See first footnote to § 1.
and Refraction of Solitary Plane Waves. 187
orientation of equilibrium, and k as the bulkmodulus, that is
to say, the reciprocal of the compressibility, of ether. Thus
we now have as before in equations (15), (11), and (18)
P • ^2. J^ — 1,2— P
a 2 + b 2 = u~ 2 =^; a 2 + b 2 = u {
a 2+ c 2 = v 2 = P_. a 2 + c 2 = v 2P
(24).
K Kj
Using (20) and (21) in (23) with y = we nod
n(a? + b 2 )(I + I') =71^ + 0?)!, J ' * [ ] '
whence by (24)
pJ'=p,J<; p(i+i')=M; (26).
By these equations eliminating \ and J, from (22), we find
{bp J b J p)l+(bp l + b l p)V=a{p l p)Z l
X,,} ■ ■ ■ m
«0»p)(I+I') = (cp
and solving these equations for 1' and J' in terms of I, w
have
jr_ (bpb^jcpt + cp) a 2 (pp) 2 j
{bp l + bfi) {cp, + c t p) + a (pp) 2 ,
j, = 2ab Pl {pp) • *
^P,+ b iP) {cpi + W) +a 2 (Pipy
and with J' and V thus determined, (26) give J y and I,,
completing the solution of our problem.
§ 13. Using (18) to eliminate a, b, b n c, and c t , from (28),
and putting
, P/ ~ P ,. =h (29);
we find
r _ p, cot i— p cot ih(pp)
I /^coU + pcot^ + A^p) ^ oU ^
and
J /_ 2/^ cot?
. . . (ol).
I p, voti + p cot ij\ h(p j— p)
the case of t? and ^ y ve
ad Uj\ which by (2S) make
cotj^l/va, and coty y = l/v y a . . . (32).
Consider now the case of v and v, very small in compa
rison with u and w y ; which by (28) makes
188 Lord Kelvin on the Reflexion
This gives
h ^ (pp) S ini (33)
r/ v r V,
which is a very small numeric. Hence J 7 is very small in
comparison with I ; and
I' ^ p, cot ip cot ij ^ (M .
1 * pi cot i + p cot i t
§14. If the rigidities of the two mediums are equal, we
have p / \p = sin 2 i \ sin 2 i n and (34) becomes
V _ sin 2i— sin 2i / _ tan (i—ij) ,„~.
I ~~ sin 2i + sin 2i y ~ tan (i + i y ) ^ v'
which is Fresnel's " tangentformula." On the other hand,
if the densities are equal, (34) becomes
I'_ —gin ft'— t,) (9C]
I~ sin(t + t,) •.•••• WW.
which is FresnePs " sineformula " ; a very surprising and
interesting result. It has long been known that for vibrations
perpendicular to the plane of the incident, reflected, and
refracted rays, unequal densities with equal rigidities of the
two mediums, whether compressible or incompressible, gives
Fresnel's sinelaw : and unequal rigidities, with equal
densities, gives his tangentlaw. But for vibrations in the
plane of the three rays, and both mediums incompressible,
unequal rigidities with equal densities give, as was shown by
Rayleigh in 1871*, a complicated formula for the reflected
ray, vanishing for two different angles of incidence, if the
motive forces in the waves are according to the law of
the elasticity of an ordinary solid. Now we find for vibrations
in the plane of the rays, Fresnel's sinelaw, with its continual
increase of reflected ray with increasing angles of incidence
up to 90°, if the restitutional forces follow the law of
dependence on rotation which I have suggested f for ether,
and if the waves of condensation and rarefaction travel at
velocities small in comparison with those of waves of dis
tortion.
§15. Interesting, however, as this may be in respect to an
ideal problem of dynamics, it seems quite unimportant in the
wavetheory of light ; because Stokes J has given, as I
* Phil. Mag. 1871, 2nd half year.
t " On the Reflexion and Refraction of light," Phil. Mag. vol. xxvi.
1888.
% "' Dynamical Theory of Diffraction." See footnote §5.
and Refraction of Solitary Plane Waves. 189
believe, irrefragable proof that in light polarized by reflexion
the vibrations are perpendicular to the plane of the incident
and reflected rays, and therefore, that it is for vibrations in
this plane that Fresnel's tangentlaw is fulfilled.
§ 16. Of our present results, it is (35) of §14 which is really
important ; inasmuch as it shows that Fresnel's tangentlaw
is fulfilled for vibrations in the plane of the rays, with the
rotational law of force, as I had found it in 1888 * with the
elasticsolidlaw of force, provided only that the propagational
velocities of condensational w T aves are small in comparison
with those of the waves of transverse vibration which
constitute light.
§17. By (28) we see that when a~ l , the velocity of the
wavetrace on the interface of the two mediums, is greater
than the greatest of the wavevelocities, each of b, b n c, Cj is
essentially real. A case of this character is represented by
fig. 2, in which the velocities of the condensational waves
in both mediums are much smaller than the velocity of the
refracted distortion al wave, and this is less than that of the
incident wave which is distortional. When one or more of
b, b /y c, Cy is imaginary, our solution (26) (28) remains valid,
but is not applicable to /regarded as an arbitrary function ;
because although f(t) may be arbitrarily given for every real
value of t, we cannot from that determine the real values of
f(t +l g)+f(t lq ) (37),
»{/(* + «?)/(««?)} .... (38).
The primary object of the present communication was to
treat this case in a manner suitable for a single incident soli
tary wave whether condensational or distortional ; instead of
in the manner initiated by Green and adopted by all subsequent
writers, in which the realized results are immediately applicable
only to cases in which the incident wavemotion consists of
an endless train of simple harmonic waves. Instead, therefore,
of making / an exponential function as Green made it, I
take
tothz < 39 )'
where r denotes an interval of time, small or large, taking the
place of the " transittime " (§7 above), which we had for the
case of a solitary wavemotion starting from rest, and coming
to rest again for any one point of the medium after an interval
of time which we denoted by t.
* See footnote §14.
190 On Reflexion and Refraction of Solitary Plane Waves.
§18. Putting now
I=p + iq (40);
and from this finding V, I n J', J / ; and taking for the real
incident wavemotion (§10 above)
f = V _ i r p + iq pig  "
b a 2 Lt—ax + by + iT t — ax + by — trj \
_ p(t — ax + by) + qT 
(tax + by)* + T 2 J
being the mean of the formulas for f 1 and —t; we find a
real solution for any case of b n c, c n some or all of them
imaginary.
§19. Two kinds of incident solitary wave are expressed by
(41), of types represented respectively by the following
elementary algebraic formulas : —
tax + by
and
{t—ax + byf + i*
(tax + byf + T*
(43).
The same formulas represent real types of condensational
waves with f/a and y/( — c), instead of the f/6 and rj/a of (41)
which relates to distortional waves. It is interesting to
examine each of these types and illustrate it by graphical
construction : and particularly to enquire into the distribution
of energy, kinetic and potential, for different times and places
in a wave. Without going into details we see immediately
that both kinetic and potential energy are very small for any
value of (t—ax + by) 2 which is large in comparison with t 2 .
I intend to return to the subject in a communication
regarding the diffraction of solitary waves, which I hope to
make at a future meeting.
§20. It is also very interesting to examine the type
formulas for disturbance in either medium derived from (41)
for reflected or refracted waves when b n or c, or c/ is
imaginary. They are as follows, for example if b ; = ty, where
g is real ;
t ^ ax ... . (44)
(tax) 2 + (gy + T) 2 { h
9y + T ... (45).
(tax)*+(gy + r)
Prof. H. L. Calleudar on Platinum Thermometry. 191
These real resultants of imaginary waves are not plane
waves. They are forced linear waves sweeping the interface,
on which they travel with velocity a~ ] ; and they produce
disturbances penetrating to but small distances into the
medium to which they belong. Their interpretation in con
nexion with total internal reflexion, both for vibrations in the
plane of the rays, and for the simpler case of vibrations
perpendicular to this plane (for which there is essentially no
condensational wave) constitutes the dynamical theory of
FresnePs rhomb for solitary waves.
XIII. Notes on Platinum Thermometry. By H. L. Cal
LEKDAK, M.A., P.P.S., Quain Professor of Physics, Uni
versity College , London*.
SINCE the date of the last communication, which I made to
this Journal in February 1892, I have been continually
engaged in the employment of platinum thermometers in
various researches. But although I have exhibited some of
my instruments at the Royal Society and elsewhere and have
described the results of some of these investigations, I have
not hitherto found time to publish in a connected form an
account of the construction and application of the instruments
themselves, or the results of my experience with regard to the
general question of platinum thermometry. As the method
has now come into very general use for scientific purposes, it
may be of advantage at the present time to collect in an
accessible form some account of the progress of the work, to
describe the more recent improvements in methods and
apparatus, and to discuss the application and limitations of
the various formulas which have from time to time been
proposed.
The present paper begins with a brief historical summary,
with the object of removing certain common misapprehensions
and of rendering the subsequent discussion intelligible. It
then proceeds to discuss various formulas and methods of
reduction, employing in this connexion a proposed standard
notation and nomenclature, which I have found convenient in
my own work. I hope in a subsequent paper to describe some
of the more recent developments and applications of the
platinum thermometer, more particularly those which have
occurred to me in the course of my own work, and which have
not as yet been published or described elsewhere.
* Communicated by the Author.
192 Prof. H. L. Callendar on Platinum Thermometry,
Historical Summary.
The earlier experiments on the variation of the electrical
resistance of metals with temperature were either too rough,
or too limited in range, to afford any satisfactory basis for a
formula. The conclusion of Lenz (1838), that the resistance
reached a maximum at a comparatively low temperature,
generally between 200° and 300° C, was derived from the
empirical formula,
R°/R=l + at'+bt 2 , (L)
in which R° and R stand for the resistances at 0° and t° C,
respectively. This conclusion resulted simply from the
accident that he expressed his results in terms of conductivity
instead of resistance, and could be disproved by the roughest
qualitative experiments at temperatures beyond the range
0° to 100° C, to which his observations were restricted.
Matthiessen (1862), in his laborious and extensive investiga
tions, also unfortunately fell into the same method of
expression. His results have been very widely quoted and
adopted, but, owing to the extreme inadequacy of the formula,
the accuracy of his work is very seriously impaired even
within the limits of the experimental range to which it was
confined. The socalled Law of Glausius, that the resistance
of pure metals varied as the absolute temperature, was a
generalization founded on similarly incomplete data. The
experiments of Arndtsen (1858), by which it was suggested,
gave, for instance, the temperaturecoefhcients '00394 for
copper, '00341 for silver and *00413 for iron, all of which
differ considerably from the required coefficient '003G65.
The observations, moreover, were not sufficiently exact to
show the deviation of the resistancevariation from lineality.
The experiments of Sir William Siemens (1870) did not
afford any evidence for the particular formula which he pro
posed, at least in the case of iron. These formulas have been
already discussed in previous communications'^, but con
sidering the extent to which they are still quoted, it may be
instructive to append the curves representing them, as a
graphic illustration of the danger of applying for purposes of
extrapolation formulas of an unsuitable type. The curves
labelled Morrisf and Benoit, which are of the same general
character but differ in steepness, may be taken as representing
approximately the resistancevariation of specimens of pure
and impure iron respectively.
The first experiments which can be said to have afforded
any satisfactory basis for a general formula were those of
* Callendar, Phil. Mag. July 1891 ; G. M. Clark, Electrician, Jan. 1897.
t Phil. Mag. Sept. 1897, p. 213.
Prof. H. L. Callendar on Platinum Thermometry. 193
Benoit (Comptes JRendus, 1873, p. 342). Though apparently
little known and seldom quoted, his results represent a great
advance on previous work in point of range and accuracy.
Fio. 1.
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Temp.
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—300° — 2JG° 100 100° 200° 300° 400 J 500° 600° 700° 300° 900° 000°
The wires on which he experimented were wound on clay
cylinders and heated in vapourbaths of steam (100°), mercury
(360°), sulphur (440°), and cadmium (860°), and in a liquid
bath of mercury for temperatures below 360°. The resistances
were measured by means of a Becquerel differential galvano
meter and a rheostat consisting of two platinum wires with a
sliding mercurycontact. It is evident that the values which
he assumed for the higher boilingpoints are somewhat rough.
The boilingpoint attributed to cadmium, following Deville
and Troost, is about 50° too high according to later experi
ments by the same authorities, or about 90° too high according
to Carnelley and Williams. It would appear also that no
special precautions were taken to eliminate errors due to
thermoelectric effects, to changes in the resistance of the
leading wires, and to defective insulation, &c. In spite of
these obvious defects it is surprising to find how closely the
results as a whole agree with the observations of subsequent
investigators. The resistancevariation of all the more common
194 Prof. H. L. "Callendar on Platinum Thermometry.
metals, according to Benoit, is approximately represented by
an empirical formula of the type
R/R°=l + at + bt 2 , (B)
where B is the resistance at any temperature t, and R° the
resistance at 0° C. The values of the constants a and b which
he gives for iron and steel represent correctly (in opposition
to the formula of Siemens) the very rapid increase in the rate
of change of resistance with temperature, as shown by the
relatively large positive value of the coefficient b. He gives
also in the case of platinum a small negative value for b (a
result since abundantly confirmed), although the specimen
which he used was evidently far from pure*. This formula,
which is the most natural to adopt for representing the
deviations from lineality in a case of this kind, had been
previously employed to a limited extent by others for the
variation of resistance with temperature; but it had not pre
viously been proved to be suitable to represent this particular
phenomenon over so extended a range.
The work of the Committee of the British Association in
1874 was mainly confined to investigating the changes of
zero of a Siemens pyrometer when heated in an ordinary fire
to moderately elevated temperatures. Finding that the pyro
meter did not satisfy the fundamental criterion of givinc
always the same indication at the same temperature, it did.
not seem worth while to pursue the method further, and the
question remained in abeyance for several years. In the
meantime great advances were made in the theory and
practice of electrical measurement, so that when I com
menced to investigate the subject at the Cavendish Labo
ratory, the home of the electrical standards, in 1885, I was
able to carry out the electrical measurements in a more
satisfactory manner, and to avoid many of the sources of
error existing in previous work. The results of my investi
gations were communicated to the Boyal Society in June
1886, and were published, with additions, in the ; Philoso
phical Transactions ' of the following year. Owing to a
personal accident, no complete abstract of this paper as a
whole was ever published ; and as the paper in its original
form is somewhat long and inaccessible, many of the points
it contained have since been overlooked. The greater part of
the paper was occupied with the discussion of methods and
observations with airthermometers ; but it may not be amiss
at the present time to give a summary of the main conclusions
* It may be remarked that the sign of this coefficient for platinum and
palladium is wrongly quoted in Wiedemann, JSlectricitat, vol. i. p. 525.
Prof. H. L. Callendnr on Platinum Thermometry. 195
■which it contained, so far as they relate to the subject of
platinum thermometry.
(1) It was shown that a platinum resistancethermometer,
if sufficiently protected from strain and contamination, was
practically free from changes of zero over a range of 0° to
1200° C, and satisfied the fundamental criterion of giving
always the same indication at the same temperature.
(2) It was proposed to use the platinum thermometer as a
secondary standard, the temperature pt on the platinum scale
being defined by the formula
^ = 100(RE°)/(R / E°), .... (1)
in which the letters R, R°, R / si and for the observed resist
ances at the temperatures pt, 0°, and 100° C. respectively.
(3) By comparing the values of ' pt deduced from different
pairs of specimens of platinum wires, wound side by side and
heated together in such a manner as to be always at the same
temperature, it was shown that different wires agreed very
closely in giving the same value of any temperature pt on the
platinum scale, although differing considerably in the values
of their temperaturecoefficients. (See below, p. 209.)
(4) A direct comparison was made between the platinum
scale and the scale of the airthermometer by means. of several
different instruments, in which the coil of platinum wire was
enclosed inside the bulb of the airthermometer itself, and so
arranged as to be always at the same mean temperature as the
mass of air under observation. As the result of this com
parison, it was shown that the small deviations of the platinum
scale from the temperature t by airthermometer could be
represented by the simple differenceformula
D = tpt = d(tl\00l)tll00, .... (2)
with a probable error of less than 1° C. over the range 0° to
650° 0.
(5) It was inferred from the comparisons of different
specimens of wire referred to in (3) (which comparisons were
independent of all the various sources of error affecting the
airthermometer, and could not have been in error by so much
as a tenth of a degree) that the simple parabolic formula did
not in all cases represent the small residual differences between
the wires.
(6) It was shown by the direct comparison of other typical
metals and alloys with platinum, that the temperaturevariation
of the resistance of metals and alloys in general could probably
be represented by the same type of formula over a consider
able range with nearly the same order of accuracy as in the
196 Prof. H. L. Callendar on Platinum Thermometry.
case of platinum. But, that the formula did not represent
singularities due to change of state or structure, such as those
occurring in the case of iron at the critical temperature, or in
the case of tin at the point of fusion.
This paper attracted very little attention until the results
were confirmed by the independent observations of Griffiths**,
who in 1890 applied the platinum thermometer to the deter
mination of certain boiling and freezingpoints, and to the
testing of mercury thermometers of limited scale. The
results of this work appeared at first to disagree materially
with the differenceformula already quoted, the discrepancy
amounting to between 6° and 7° at 440° C. After his work
had been communicated to the Royal Society a direct com
parison was made with one of my thermometers in his appa
ratus ; and the discrepancy was traced to the assumption by
Griffiths of RegnaiuYs value 448°'38 C. for the boilingpoint
of sulphur. We therefore undertook a joint redetermination
of this point with great care, employing for the purpose one
of my original airthermometers which had been used in the
experiments of 1886. The results of this determination were
communicated to the Royal Society in December 1890, and
brought the observations of Griffiths into complete harmony
with my own and with the most accurate work of previous
observers on the other boiling and freezingpoints in question.
The agreement between his thermometers when reduced by
the differenceformula (2), employing for each instrument the
appropriate value for the differencecoefficient d, was in fact
closer than I had previously obtained with platinum wires
from different sources. But the agreement served only to
confirm the convenience of the method of reduction by
means of the Sulphur BoilingPoint (S.B.P.) which we
proposed in that paper ".
Proposed Standard Notation and Nomenclature.
It will be convenient at this stage, before proceeding to
discuss the results of later work, to explain in detail the
notation and phraseology which I have found to be useful in
connexion with platinum thermometry. This notation has
already in part been adopted by the majority of workers
in the platinum scale, and it would be a great saving in time
and space if some standard system of the kind could be gene
rally recognized. In devising the notation special attention
has been paid to the limitations of the commercial typewriter,
as the majority of communications to scientific societies at
* Phil. Trans, clxxxii. (1891), A, pp. 4S72.
f Ibid. t.c. pp. 119157.
Prof. H. L. Callendar on Platinum Thermometry. 197
Ihe present time are required to be typewritten. It is for
this reason desirable to avoid, wherever possible, the use of
Greek letters and subscript diacritics and indices.
The Fundamental Interval. — The denominator, B/ — R°, in
formula (1) for the platinum temperature pt, represents the
change of resistance of the thermometer between 0° and
100° C, and is called the fundamental interval of the thermo
meter, in accordance with ordinary usage. It is convenient,
as suggested in a previous communication, to adjust the
resistance of each thermometer, and to measure it in terms
of a unit such that the fundamental interval is approxi
mately 100. The reading of the instrument will then give
directly the value of pt at any temperature, subject only to a
small percentage correction for the error of adjustment of the
fundamental interval.
The Fundamental Coefficient. — The mean value of the tem
peraturecoefficient of the change of resistance between 0°
and 100° C. is called the fundamental coefficient of the wire,
and is denoted by the letter c. The value of c is given by the
expression (B/  R°)/100 R°. The value of this coefficient is
not necessary for calculating or reducing platinum tempe
ratures, but it is useful for identifying the wire and as giving
an indication of its probable purity.
The Fundamental Zero. — The reciprocal of the fundamental
coefficient c is called the fundamental zero of the scale of the
thermometer, and. is denoted by the symbol pt°, so that
pt°=l/c. The fundamental zero, taken with the negative
sign, represents the temperature on the scale of the instrument
itself at which its resistance would vanish. It does not
necessarily possess any physical meaning, but it is often more
convenient to use than the fundamental coefficient (e. g. y
Phil. Trans. A, 1887, p. 225). It may be remarked that,
if the resistance has been accurately adjusted so that the
fundamental interval is 100 units, R°, the resistance at 0° C,
will be numerically equal to pt°.
The Difference Formula. — It is convenient to write the
formula for the difference between t and pt in the form already
given (2), as the product of three factors, d x (£/100— 1) x t/\ 00,
rather than in the form involving the square of £/100, which I
originally gave, and which has always been quoted. Owing
to the form in which it was originally cast, I find that most
observers have acquired the habit of working the formula in
the following manner. First find the square of f/100,
then subtract £/100, writing the figures down on paper, and
finally multiply the difference by the differencecoefficient d
with the aid of a sliderule. It is very much easier to work
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. P
198 Prof. H. L. Callendar on Platinum Thermometry.
the formula as the product of three factors, because the sub
traction (7/100 — 1) can be safely performed by mental
arithmetic. The whole process can then be performed by
one application of the sliderule, instead of two, and it is
unnecessary to write down any intermediate steps on paper.
The Parabolic Function. — It is convenient to have an
abbreviation for the parabolic function of t vanishing at
0° and 100°, which occurs so frequently in questions of
thermometry. I have found the abbreviation p(t) both sug
gestive and useful for this purpose. The formula may then
be written in the abbreviated shape, t=pt\dxp(t).
The "S.B.P." Method of Reduction.— Assuming the differ
enceformula, the value of the differencecoefficient d may be
determined by observing the resistance B", and calculating
the corresponding value of the platinum temperature pt" . at
some secondary fixed point t n , the temperature of which is
known on the scale of the gasthermometer. The boiling
point of sulphur (S.B.P.) is generally the most convenient
to use, and has been widely adopted for this purpose. As
suming that this point is chosen for the purpose, and that the
height of the barometer at the time is 760 + A millims., the
corresponding temperature is given by the formula
t" =44453 + 082 h,
provided that h is small, and the corresponding value of the
parabolic function by the formula
p(t") = 15*32 + 0065 h,
whence d=(t"pt // )[p(t // ).
With the best apparatus it is possible to attain an order of
accuracy of about 0*1 per cent, in the value of d obtained by
this method, at least in the case of thermometers which are
not used at temperatures above 500°. At higher temperatures
the exact application of the formula would be more open to
question, and it may be doubted whether the value of the
differencecoefficient would remain constant to so small a
fraction of itself.
Other Secondary Fixed Points. — For very accurate work
between 0° and 100° C. it might be preferable to use a value
of d determined at 50° C, either by direct comparison with
an airthermometer or by comparison with a standard platinum
thermometer. The latter comparison would be much the
easier and more accurate. Although the most careful com
parisons have hitherto failed to show that the value of d
obtained by assuming the S.B.P. does not give correct results
Prof. H. L. Callendar on Platinum Thermometry. 199
between 0° and 100° C, it is quite possible that this might not
always be the case.
For work at low temperatures it would be preferable, from
every point of view, to make use of the boilingpoint of oxygen
as the secondary fixed point. There appears to be a very
general consensus of opinion that the temperature of liquid
oxygen boiling under a pressure of 760 mm. is — 182°*5 C,
on the scale of the constantvolume hydrogen or helium
thermometer*. It is quite possible that, as in the case of
water and most other liquids, the temperature of the boiling
liquid would be different from that of the condensing vapour
at the same pressure ; but the boiling liquid is the most con
venient to employ, and it appears that its temperature is
steady to two or three tenths of a degree, and reproducible by
different observers to a similar order of accuracy. I have
found it convenient for purposes of distinction to employ the
symbol d° to denote the value of d deduced from the boiling
point of oxygen, and the symbol d" to denote that deduced
from the boilingpoint of sulphur. The formulae for the
pressure correction in the case of oxygen are approximately
t= 1825 + 020 A; p(t) = 51600093A.
The Resistance Formula. — I have shown in the paper
already referred to that the adoption of the parabolic differ
enceformula for the relation between pt and t is equivalent
to assuming for the resistance variation the formula
R,/B° = l + at + bt* (3)
The values of the coefficients a and b are found in terms of c
and d, or vice versa, by means of the relations
a=c {l + d/100), b=cd/10fl00.
Graphic Method of Reduction. — The quickest and most
generally convenient method of reducing platinum tempe
ratures to the air scale is to plot the difference t— pt in terms
of t as abscissa, and to deduce graphically the curve of differ
ence in terms of pt as abscissa, as described and illustrated
in my original paper. This method is particularly suitable at
temperatures up to 500° C, as the difference over this range
is relatively small and accurately known. It is also very con
venient if a large number of determinations are to be made
with a single instrument. It is not so convenient in the case
of a number of different instruments with different coefficients,
* The experimental evidence for this number is not quite satisfactory,
owing to differences in the atmospheric pressure and impurities in the
oxygen. It must be understood that the adoption of this value is
provisional and subject to correction.
P2
200 Prof. H. L. Callendar on Platinum Thermometry.
each of which is used for a comparatively limited number of
determinations. In such a case the trouble of drawing the
separate curves, with sufficient care to be of use, would more
than counterbalance the advantage to be gained by the
method.
Hey cock and Neville's Method. — In order to avoid this
difficulty Messrs. Heycock and Neville, in their classical
researches at high temperatures * devised an ingenious modi
fication of procedure, which has given very good results
in their hands, but is not quite identical with the simple
differenceformula. They described a differencecurve in the
usual manner, giving the value of the difference in terms of
ft as abscissa for a standard value rf = l*50 of the difference
coefficient. The appropriate values of d were determined in
the case of each pyrometer by the S.B.P. method. In re
ducing the observations for any given values of pt and <7, the
value of the difference corresponding to pt was taken from
the curve for J = l*50, and was then multiplied by the factor
d/1'50 and added to pt. This method is very expeditious
and convenient, and gives results which are in practical
agreement with the pure differenceformula, provided that,
as was almost invariably the case in their observations, the
values of d do not differ materially from the average 1*50.
If, however, the pure differenceformula is correct, the method
could not be applied in the case of values of d differing con
siderably from the average. The difference between the
methods cannot be simply expressed in terms of either pt or t
for considerable variations in the value of d. But for a small
variation Sd in the value of d in the vicinity of the normal
value, it is easy to show that the difference St between
the true value of t as given by the differenceformula
t—pt=.dp(t), and the value found by the method of Heycock
and Neville, is approximately
8t=8d(dt/dptl)p(t)t.
Neglecting the variation of d entirely, the error would be
B r t=Sd(dt/dpt)p{t).
For example, at £ = 1000°, p(t)=90, (d£/dpf) = l40, we
should find for a variation of d from 1*50 to I'60, the values
& = 3°8 (H. & N.), and 8't =12°'8 (variation neglected).
This is an extreme case. In the observations of Heycock
and Neville, the values found for the coefficient d seldom
* Trans. Chem. Soc. Feb. 1895, p. 162.
t The value of dt/dpt at any point is readily found by differentiating 1
the differenceformula (2), dpt/dt=l(t/50 ljd/100.
Prof. H. L. Callendar on Platinum Thermometry. 201
varied so much as # 04 on either side of the mean, in the case
of their standard wire. It is, moreover, quite possible that
these variations may have been partly due to fortuitous differ
ences at the S.B.P. and at the fixed points, in which case it
is probable that the Heycock and Neville method of reduction
would lead to more consistent results than the pure difference
formula, because it does not allow full weight to the apparent
variations of d as determined by the S.B.P. observations.
It is clearly necessary, as Heycock and Neville have shown,
and as the above calculation would indicate, to take some
account of the small variations of d, at least in the case of
pyrometers in constant use at high temperatures. The method
of Heycock and Neville appears to be a very convenient and
practical way of doing this, provided that the variations of d
are small. It must also be observed that, although the indi
vidual reductions by their method may differ by as much as
1° or 2° at 1000° from the application of the pure difference
formula, the average results for the normal value of d will be
in exact agreement with it.
Difference Formula in Terms of pt. — In discussing the
variation of resistance as a function of the temperature, it is
most natural and convenient to express the results in terms of
the temperature t on the scale of the airthermometer by
means of the parabolic formula already given. This formula
has the advantage of leading to simple relations between the
temperaturecoefficients ; and it also appears to represent the
general phenomenon of the resistancevariation of metals
over a wide range of temperature with greater accuracy than
any other equally convenient formula. When, however, it
is simply a question of finding the temperature from the
observed value of the resistance^ or from the observed reading
of a platinum thermometer, over a comparatively limited
range, it is equally natural, and in some respects more con
venientj to have a formula which gives t directly in terms of
pt or R. This method of expression was originally adopted
by Griffiths, who expressed the results of the calibration of
his thermometers by means of a formula of the type
t—pt = apt + bpt 2 + cpt d + dpt 4 . ...(G)
The introduction of the third and fourth powers of pt in
this equation was due to the assumption of RegnauhVs value
for the boilingpoint of sulphur. If we make a correction
for this, the observations can be very fairly represented by a
parabolic formula of the type already given, namely,
tpt=dXpt/100l)pt/l0Q=d / p(pty . . (I)
202 Prof. H. L. Callendar on Platinum Thermometry.
This formula is so simple and convenient, and agrees so
closely over moderate ranges of temperature with the ordinary
differenceformula, as to be well worth discussion. I have
been in the habit of using it myself for a number of years in
approximate reductions at moderate temperatures, more par
ticularly in steamengine and conductivity experiments, in
which for other reasons a high degree of accuracy is not
required. It has also been recently suggested by Dickson
(Phil. Mag., Dec. 1897), though his suggestion is coupled
with a protest against platinum temperatures.
The value of the difference coefficient oV in this formula
may be determined as usual by reference to the boilingpoint
of sulphur, or it may be deduced approximately from the
value of the ordinary differencecoefficient d by means of the
relation
d'=d/(l077d), or d=d'/{l + '077d').
If this value is chosen for
enceformulae will of course
the coefficient, the two dii
at 0°, 100 c
and 445° C,
The order of
but will differ slightly at other temperatures,
agreement between the formulae is shown at various points of
the scale by the annexed table, in which t represents the
temperature given by the ordinary formula t—pt=l'50p(t),
and t' the temperature calculated by formula (4) for the same
value of pt, choosing the value d'= 1*695, to make the two
formulae agree at the S.B.P.
Table I.
Comparison of DifferenceFormulae, (2) & (4).
t
300°
4° 5
200°
l°95
100°
0°54
+•50°
+•050°
200°
•23°
300° !
•42°
tv ...
t
400°
•25°
600°
+2°2
800°
+9°3
1000°
+22° 9
1200°
+46°6
1500°
+97°'2
tt' ...
It will be observed that the difference is reasonably small
between the limits — 200° and + 600°, but that it becomes
considerable at high temperatures. A much closer agreement
may be readily obtained over small ranges of temperature by
choosing a suitable value of d' . The two formulae become
practically indistinguishable between 0° and 100°, for in
stance, if we make d f = d. For steamengine work I generally
selected the value of d f to make the formulae agree at 200° C.
Prof. H. L. Callendar on Platinum Thermometry. 203
For work at low temperatures, it would be most conveuient to
select the boilingpoint of oxygen for the determination of
either differencecoefficient. The two formulae are so similar
that they cannot be distinguished with certainty over a
moderate range of temperature. But if the values of the differ
encecoefficients are calculated from the S.B.P., the balance
of evidence appears to be in favour of the original formula (2).
Formula (4) appears to give differences which are too large
between 0° and 100° C; and it does not agree nearly so
well as (2) with my own air thermometer observations over
the range 0° to 650° C. It appears also from the work of
Heycock and Neville to give results which are too low at high
temperatures as compared with those of other observers.
It is obvious, from the similarity of form, that the differ
enceformula (4) in terms of pt corresponds, as in the case of
formula (2), to a parabolic relation between the temperature
and resistance, of the type
^^Ha'R/^ + Z/CR/R^^a'XR/^lJ+^^R/R !) 2 . (5)
When R = 0, t =  t° = (<*"&"). Also V= b", and
a'=a"2&".
The values of the fundamental coefficient c, and of the fun
damental zero pt°, are of course the same on either formula,
provided that they are calculated from observations at 0° and
100° C, but not, if they are calculated from observations
outside that range. The values of the coefficients a" and h"
are given in terms of d' ', and either pt°, or c, by the relations
a" =pt° (1  d'/lOO) = ( 1  d7 100) /c, and d! = 10,0006V.
Formulae of this general type, but expressed in a slightly
different shape, have been used by Holborn and Wien for
their observations at low temperatures, and recently by
Dickson for reducing the results of Fleming and other
observers. But they do not employ the platinum scale or
the differenceformula.
Maximum and Minimum Values of the Resistance and
Temperature, — It may be of interest to remark that the dif
ferenceformulae (2) and (4) lead to maximum or minimum
values of pt and t respectively, which are always the same
for the same value of d, but lie in general outside the range
of possible extrapolation. In the case of formula (2), the
resistance reaches a maximum at a temperature t= —a/'lb —
(5000/d) (l + d/100). The maximum values of pt and R are
given in terms of d and c by the equations
pt (max.) = (1 + d/100) t/2 = (2500/rf) (1 + d/lOO)*,
R/R° (max.) =l+pt (max.)//rf?=l+ (2500c/d) (l+^/lOO) 2 .
204 Prof. H. L. Gallendar on Platinum Thermometry.
Similarly in the case of the differenceformula (4) in terms
of pt, the maximum or minimum value of t is given in terms
of d' by the equation
*(max.) = (ldyiOOXp*/2= (2500/d') (ltf'/lOO) 2 .
Dickson's Formula. — In a recent number of this Journal
(Phil. Mag., Dec. 1897) Mr. Dickson has proposed the formula
(R+ a y= P (t+b) (6)
He objects to the usual formula (3) on the grounds, (1)
that it leads to a maximum value of the resistance in the
case of platinum at a temperature of about £ = 3250° C, and
(2) that any given value of the resistance corresponds to two
temperatures. He asserts that " both of these statements
indicate physical conditions which we have no reason to sup
pose exist/' In support of contention (1), he adduces a rough
observation of Holborn and Wien * to the effect that the
* Wied. Ann. Oct. 1895 ; p. 386. Mr. Dickson and some other writers
appear to attach, too much weight to these observations of Messrs. Holborn
and Wien. So far as they go, they afford a very fair confirmation of the
fundamental principles of platinum thermometry at high temperatures ;
but the experiments themselves were of an incidental character, and
were made with somewhat unsuitable apparatus. Only two samples of
wire were tested, and the resistances employed were too small for
accurate measurement. The wires were heated in a badlyconducting
inutile and were insulated by capillary tubes of porcelain or similar
material. The temperature of the wire under test was assumed to be the
mean of the temperatures indicated by two thermojunctions at its
extremities ; but the authors state that " the distribution of temperature
in the furnace was very irregular." The resistance was measured by a
modification of the potentiometer method, and no attempt was made to
eliminate residual thermoelectric effects. Under these conditions the
observations showed that the resistance was not permanently changed by
exposure to a temperature of 1600° C, at least within the limits of
accuracy of the resistance measurements. It is quite easy, however, by
electric heating as in the " nieldonreter," to verify the differenceformula
at high temperatures, with less risk of strain or contamination or bad
insulation. (See Petavel, Phil. Trans. A (1898), p. 501.)
The two series of observations (excluding the series in which the tube
of the muffle cracked, and the thermocouples and wire were so con
taminated with silicon and furnacegases as to render the observations
valueless) overlapped from 1050° to 1250° C, and showed differences
between the two wires varying from 10° to 45° at these temperatures,
the errors of individual observations in either series being about 10° to
15°. It must be remembered, however, that the two wires were of
different sizes and resistances ; they were heated in different furnaces ;
they were insulated with different materials; and their temperatures
were deduced from different thermocouples. Taking these facts into
consideration, it is remarkable that the observed agreement should be so
close. The observations at the highest temperatures in both cases, with
the furnaces full blast and under the most favourable conditions for
securing uniformity of temperature throughout the length of the wire, are
in very close agreement with the differenceformula (2), assuming d=l75.
The second specimen was also tested at lower temperatures, but the
Prof. H. L. Callendar on Platinum Thermometry. 205
resistance of one of their wires had already nearly reached
6R° at a temperature of about l(U0 n C, whereas the maximum
calculated resistance in the case of one of my wires (with a coeffi
cient c= '00340) was only (r576R°. He omits to notice that
the result depends on the coefficients of the wire.
The wire used by Holborn and Wien had a fundamental
coefficient c = '00380, and the highest value of the resistance
actually observed was not 6R° as suggested, but B/R° — 5'53,
at a temperature £ = 1610° C, deduced from thermoj unctions
at each end. If we assume d=l'70 as a probable value of
the differencecoefficient for their wire, the differenceformula
(2) would give,
at £=1610°, D = 414°, ^=1196°, whence R/R° = 5'54.
It w T ould be absurd to attach much weight to so rough an
observation, but it will be seen that, so far as it goes, the
result is consistent with the usual formula, and does not bear
out Mr. Dickson's contention. A more important defect in
arguments (1) and (2) lies in the fact that maximum and
minimum values of the resistance are known to occur in
the case of manganin and bismuth within the experimental
range, and that such cases can be at least approximately
represented by a formula of the type (3), but cannot be
represented by a formula of the type (6).
As shown by Table I. above, the formula proposed by
Dickson agrees fairly well with formula (3), in the special
case of platinum, through a considerable range. Rut the
case of platinum is exceptional. If we attempt to apply a
formula of Dickson's type to the case of other metals, we are
met by practical difficulties of a serious character, and are
driven to conclude that the claim that it is 6l more represen
tative of the connexion between temperature and resistance
than any formula hitherto proposed/' cannot be maintained.
observations are somewhat inconsistent, and lead to values of d which
are rather large and variable, ranging from 3:7 to 20. These variations
are probably due to errors of observation or reduction. This is shown
by the work of Mr. Tory (B.A. Report, 1897), who made a direct com
parison between the Pt — PtRh thermocouple and the platinumthermo
meter by a much more accurate method than that of Holborn and Wien.
He found the parabolic differenceformula for the platinum thermometer
to be in very fair agreement between 100° and 800° C. with the previous
series of observations of Holborn and Wien on this thermocouple (Wied.
Ann. 1892), and there can be little doubt that the discrepancies shown
by their later tests were due chiefly to the many obvious defects of the
method. For a more detailed criticism of these observations, the reader
should refer to a letter by Griffiths in ' Nature,' Feb. 27th, 1896. It is
sufficient to state here that the conclusions which these observers drew
from their experiments are not justified by the observations themselves.
206 Prof. H. L. Callendar on Platinum Thermometry.
If, for instance, we take the observations of Fleming on very
pure iron between 0° and 200° C, and calculate a formula of
the Holborn and Wien, or Dickson, type to represent them,
we arrive at a curve similar to that shown in fig. 1 (p. 193).
(The values of the specific resistance of Fleming's wire are
reduced, for the sake of comparison, to the value R = 10,000
at 0° C.) This curve agrees very closely with that of Morris
and other observers between 0° and 200° C. The peculiarities
of the curve beyond this range are not due to errors in the
data, but to the unsuitable nature of the formula. A similar
result would be obtained in the case of iron by employing
any other sufficiently accurate data. It will be observed that
the formula leads to a maximum value of the temperature
£ = 334°, and makes the resistance vanish at —197°. Below
334° there are two values of the resistance for each value of
the temperature, and the value of dR/d£ at 334° is infinite,
both of which conditions are at present unknown in the case
of any metal, and are certainly not true in the case of
iron. If, instead of taking the value observed at + 196°*1 C,
we take the value obtained at the O.B.P. to calculate the
formula, we should find a better agreement with observation
at low temperatures, but the disagreement at higher tempe
ratures would be greater.
If, on the other hand, we take the same observations, namely,
c = '00625, and R/B,°= 2*372 when * = 196°'l, and calculate a
differenceformula of the type (2) corresponding to (3), we
find a 7 =125, a = 005467, b = 000,007825. The points
marked in fig. 1 are calculated from this formula, and are
seen to be in practical agreement with the observations of
Morris up to 800°. As this formula stands the test of extra
polation so much better than that of Holborn and Wien or
Dickson, we are justified in regarding it as being probably
more representative of the connexion between resistance and
temperature.
Advantages of the Difference Formula. — Mr. Dickson's ob
jections to the platinum scale and to the differenceformula
appear to result from want of familiarity with the practical
use of the instrument. But as his remarks on this subject
are calculated to mislead others, it may be well to explain
briefly the advantages of the method, which was originally
devised with the object of saving the labour of reduction
involved in the use of ordinary empirical formulae, and of
rendering the results of observations with different instruments
directly and simply comparable.
(1) In the first place, a properly constructed and adjusted
platinum thermometer reads directly in degrees of temperature
Prof. H. L. Callendar on Platinum Thermometry. 207
on the platinum scale, just like a mercury thermometer, or
any other instrument intended for practical use. The quantity
directly observed is not the resistance in ohms, but the tem
perature on the platinum scale, either pt, or j)t+pt°. The
advantage of this method is that the indications of different
instruments become directly comparable, and that the values
of pt for different wires agree very closely. If this method
is not adopted, the resistances in ohms of different instruments
at different temperatures form a series of meaningless figures,
which cannot be interpreted without troublesome reductions.
(2) The second advantage of the differenceformula lies in
the fact that the difference is small, more especially at mode
rate temperatures, and can be at once obtained from a curve
or a table, or calculated on a small sliderule, without the
necessity of minute accuracy of interpolation or calculation.
In many cases, owing to the smallness of the difference
between the scales, the results of a series of observations
can be worked out entirely in terms of the platinum scale,
and no reduction need be made until the end of the series.
For instance, in an elaborate series of experiments on the
variation of the specific heat of water between 0° and 100° C,
on which I have been recently engaged, by a method de
scribed in the Brit. Assoc. "Report, 1897, all the observations
are worked out in terms of the platinum scale, and the re
duction to the airscale can be performed by the aid of the
differenceformula in half an hour at the end of the whole
series. As all the readings of temperature have to be taken
and corrected to the tenthousandth part of a degree, and as
the whole series comprises about 100,000 observations, it is
clear that the labour involved in Mr. Dickson's method of
reduction would have been quite prohibitive. It is only by
the general introduction of the method of small corrections
that such work becomes practicable.
On the Method of Least Squares. — There appears to be a
widespread tendency among nonmathematical observers to
regard with almost superstitious reverence the value of results
obtained by the method of least squares. This reverence in
many cases is entirely misplaced, and the method itself, as
commonly applied, very often leads to erroneous results. For
instance, in a series of observations extending over a con
siderable range of temperature, it would be incorrect to attach
equal weight to all the results, because all the sources of
error increase considerably as we depart further from the
fixed points of the scale. In a series of airthermometer
observations, the fixed points themselves stand in quite a
different category to the remainder of the observations. The
208 Prof. H. L. Callendar on Platinum Thermometry.
temperature is accurately known by definition, and is not
dependent on uncertain errors of the instrument. It is a
mistake, therefore, in reducing a series of observations of this
kind, to put all the observations, including the fixed points,
on the same footing, and then apply the method of least
squares, as Mr. Dickson has applied it in his reduction of the
results of various observers with platinum thermometers. For
instance, in order to make his formula fit my observations at
higher temperatures, he is compelled to admit an error of no
less than o, 80 on the fundamental interval itself, which is
quite out of the question, the probable error of observation on
this interval being of the order of o, 01 only. The correct
way of treating the observations would be to calculate the
values at the fixed points separately, and to use the remainder
of the observations for calculating the differencecoefficient.
Even here the graphic method is preferable to that of least
squares, because it is not easy to decide on the appropriate
weights to be attached to the different observations. Cor
recting the method of calculation in this manner, we should
find a series of differences between my observations and
Dickson's formula, of the order shown in Table I. It would
be at once obvious that the deA^ations from (6) were of a
systematic type, and that it did not represent the results of
this series of observations so well as that which I proposed.
The deviations shown in Dickson's own table are of a syste
matic character ; but they would have been larger if he had
treated the fixed points correctly.
Limitations of the DifferenceFormula. — The observations
of Messrs. Haycock and Neville at high temperatures may be
taken as showing that the simple parabolic differenceformula,
in which the value of d is determined by means of the S.B.P.
method, gives very satisfactory results, in spite of the severe
extrapolation to which it is thus subjected, provided that the
wire employed is of pure and uniform quality. If, however,
the S.B.P. method of reduction is applied in the case of impure
wires at high temperatures, it may lead to differences which
are larger than the original differences in the values of pt
before reduction. For instance, I made a number of pyro
meters some years ago with a sample of wdre having the
coefficients c = '00320, d // = l'7b. My observations on the
freezingpoints of silver and gold (Phil. Mag., Feb. 1892)
were made with some of these pyrometers. All these instru
ments gave very consistent results, but they could not be
brought into exact agreement with those constructed of purer
wire by the simple S.B.P. method of reduction, employ
ing either differenceformula (2) or (4). This is not at all
Prof. H. L. Callendar on Platinum Thermometry. 209
surprising when we consider the very large difference in the
fundamental coefficient c, which is approximately '00390 in
the case of the purest obtainable wire. The remarkable fact
is that, as stated in my original paper (see above, p. 195), the
values of pt for such different specimens of wire should show
so close an agreement through so wide a range. The differ
ence in the fundamental coefficients in this extreme case is about
20 per cent.; but the values oipt for the two wires differ by only
4° at the S.B.P., and this difference, instead of increasing in
proportion to the square of the temperature, remains of the
same order, or nearly so, at the freezingpoints of silver and
gold. Thus the wire c = '00320 gave p£ = 830° at the
Ag. F.P., but I shortly afterwards obtained with a specimen
of very pure wire (c = '003897), the value pt = 835° for the
same point. Messrs. Reycoek and Neville, using the same
pure wire, have confirmed this value. They also find for the
F.P. of gold, with different instruments, constructed of the
same wire, the average value p£ = 905 o, 8. I did not test this
point with the pure wire, but the value found by Messrs.
Heycock and Neville may be compared with the value
^ = 902°'3 (Phil. Mag., Feb. 1892), which I found at the
Mint with one of the old instruments.
From these and other comparisons of the platinum scales of
different wires, it appears likely that the deviation of the
impure wire from the parabolic curve is generally of this
nature. As shown by the comparison curves in my original
paper, the deviation follows approximately the parabolic law
up to 400°, beyond that point the curves tend to become
parallel, and at higher temperatures they often show a
tendency to approach each other again. The application of
the S.B.P. method of reduction to impure wires at high
temperatures will therefore give results which are too high,
because the value of d is calculated from the S.B.P., where
the difference between the wires is nearly a maximum. Thus,
taking the values of d from the S.B.P. for the two specimens
of wire above quoted, we find, calculating the values of t for
the Ag.F.P., and Au.F.P. from the data,
Impure wire, c = '00320, d= 1*751 ;
Ag. F.P., £ = 98P6 ; Au. F.P., £ = 1092'0.
Pure wire, c = '00390, d = l'520 ;
Ag. F.P., £ = 960'7 : Au. F.P., £=1060'7.
The results for the impure wire obtained by the S.B.P. method
of reduction are not so high as those found by Bams with a
PtPtlr thermoelement, which he compared with an air
thermometer up to 1050°. There can be little doubt.
210 Prof. H. L. Callendar on Platinum Thermometry.
however, that they are too high, and that the results given
"by the pure wire are the more probable. The latter are
approximately a mean between the values of Violle 954°, and
Holborn and Wien 971°, and may be taken, in the pre
sent state of the science of hightemperature measurement,
to be at least as probable as any other values, in spite of
the extrapolation from 445°, by which they are obtained.
The extrapolation is not really so unreasonable as many
observers seem to think. The parabolic formula for resistance
variation has been verified for a great variety of cases, through
a very wide range, and with much greater accuracy than in
the case of many socalled laws of nature. For instance, a
similar formula, proposed by Tait and Avernarius, is often
regarded as the law of the thermocouple, but the deviations
of thermocouples from this law are far wider than those of the
most impure platinum thermometer. If we take a PtPtRh
thermocouple, and apply the S.B.P. method of reduction in
the same manner as in the case of a platinum thermometer,
taking the data, t= 100°, e=650 microvolts; £ = 445°, e =
2630 mv. ; we should find d= 7*4. At *=1000°C., e=9550
mv., the temperature on the scale of the thermocouple is
^ = 1470°. The temperature calculated by the parabolic
formula is £ = 804°. Whence it will be seen that the devia
tion from the formula is about ten times as great as in the
case of a very impure platinum wire. A cubic formula was
employed by Holborn and Wien to represent their observations
at hioh temperatures with this thermocouple, but even this
formula differs by more than 20° from their observations at
150° C. It is, moreover, so unsatisfactory for extrapolation
that they preferred to adopt a rectilinear formula for deducing
temperatures above 1200° C.
There are, however, more serious objections to the adoption
of the thermocouple, except to a limited extent, as a secondary
standard: — '(1) The scale of the thermocouple is seriously
affected, as shown by the observations of Holborn and Wien
and Barus at high temperatures, and of Fleming at low
temperatures, by variations in the quality of the platinum
wire and in the composition of the alloy. (2) The sen
sitiveness of the Pt — PtEh thermocouple at moderate
temperatures is too small to permit of the attainment of
the order of accuracy generally required in standard work.
(3) No satisfactory method has yet been devised in the case
of the thermocouple for eliminating residual thermal effects
in other parts of the circuit, which materially limit* the
* My present assistants, Prof. A. W. Porter, B.Sc, and Mr. N. Eumor
fopoulos, B.Sc, whose work on Emissivity and Thermal Conductivity has
already in part been published in this Journal, employed this thermo
Prof. H. L. Callendar on Platinum Thermometry. 211
attainable accuracy. In the case of the platinum thermo
meter these effects are relatively much smaller, owing to the
large change of resistance with temperature, and can be
completely eliminated in a very simple manner.
Ag. F.P. Method of Reduction for Impure Wires. — The
simplest method of reduction for such wires at high tempera
tures, would be to take the Ag. F.P. as a secondary fixed
point instead of the S.B.P. for the determination of the
differencecoefficient d. This would in general lead to a very
close agreement at temperatures between 800° and 1200°C,
but would leave residual errors of 3° or 4° at temperatures in
the neighbourhood of the S.B.P. To obtain a continuous
formula giving results consistent to within less than 1°
throughout the range, it would be necessary to adopt the
method which I suggested in my last communication (Phil.
Mag., Feb. 1892), assuming d to be a linear function of the
temperature of the form a\bt, and calculating the values of
a and b to make the instrument agree with the pure wire at
both the S.B.P. and the Ag. F.P., taking the latter as 960°'7.
We should find for the wire (c = "00320) above quoted, d=
1580 at the Ag. F.P. If we apply this value at the Au. F.P.,
we should find £ = 1063 o, 0. But if we employ the second
method, and calculate a linear formula for d to make the
results agree throughout the scale, taking d= 1*751 at the
S.B.P., we obtain d=a+ fa = T898 — 000331*. Hence
the appropriate value of d to use at the Au. F.P. would be
d= 1;547, giving for the Au. F.P. £ = 1060°0, which is in
closer agreement with the value 1060 o, 7 given by the pure
wire. This method has also the advantage that it gives
practically perfect agreement at the S.B.P., and at all points
between 0° and 1000°. In the case of the mercury thermo
meter,, or the thermocouple, a similar cubic formula is required
to give an equally good agreement between 0° and 200° 0.
In the original paper in which the suggestion was made, I
couple very extensively in their investigations. They inform me that
they were compelled to abandon the method shortly before my appoint
ment, because in spite of every precaution which their experience could
suggest they found it impossible, owing to these residual thermal effects,
to effect a sufficiently accurate calibration of the PtPtRh thermo
couple at temperatures between 0° and 100° C. The substitution of
baser metals such as iron and germansilver at low temperatures would
no doubt partly meet this difficulty, but would involve the abandonment
of the wide range and constancy and uniformity of scale characteristic of
the platinum metals, which are qualifications so essential for a standard.
We conclude on these grounds that the application of this thermo
couple is limited to high temperatures, and tnat the contention that it is
preferable to the platinum thermometer as a secondary standard cannot
be maintained.
212 Prof. H. L. Callendar on Platinum Thermometry.
assumed tentatively a much lower yalue £=945° for the Ag.
F.P., giving a result £=1037° for the Au. F.P., which
naturally does not agree with the results of subsequent work.
These results have since been misquoted in a manner which
has the effect of suggesting that the platinum thermometer
gives very capricious results at high temperatures. Holborn
and Wien, for instance, quote my value 981°'6 for the Ag.
F.P., obtained with the impure wire by the S.B.P. method
of reduction, and at the same time quote the value 1037°
for the Au. F.P., which was obtained by assuming the value
945° for the Ag. F.P. Comparing these with the values
obtained by Heycock and Neville with the pure w T ire, one
might naturally conclude, in the absence of information as to
the manner in which the two results were calculated, that
different wires gave very inconsistent results. The truth is, on
the contrary, that very different wires agree with remarkable
uniformity in giving approximately the same platinumseal e,
and that they also give consistent values of t provided that the
reduction is effected in a consistent manner. But, although
it is evident that this method may be made to give consistent
results in the case of impure wires, it is in all cases preferable
to use pure wire of uniform quality. If, forinstancej a pyro
meter gives a value of c less than '0035, or a value of d
greater than 1*70, it would be safer to reject it, although it
may possibly give very consistent results. Values of d greater
than 2*00 at the S.B.P. sometimes occur, but may generally
be taken as implying that the wire is contaminated. Such
instruments as a rule deteriorate rapidly, and do not give
consistent results at high temperatures.
The Difference Formula at Loiv Temperatures. — The suita
bility of the Platinum thermometer as an instrument for low
temperature research is shown by the work of Dewar and
Fleming, and Olszewski. It has also been adopted by Holborn
and Wien, in spite of their original prejudice against the
instrument. The first verification of the platinum scale at
very low temperatures was given by Dewar and Fleming,
whose researches by this method are the most extensive and
important. They found that two different specimens of wire
with fundamental coefficients c = "00353, and c = "00367
respectively, agreed very closely in giving the same values of
the platinum temperature down to —220°. The values of
the differencecoefficients for these wires, calculated by
assuming t= — 182°*5 for the boilingpoint of liquid oxygen,
are d = 2'15 * and d = 2'72, respectively. The first of these
refers to the particular wire which Dewar and Fleming
selected as their standard.
* See below, p. 219, middle, and footnote.
Prof. H. L. Callendar on Platinum Thermometry. 213
As an illustration of the method of reduction by the differ
enceformula, it may be of interest to reproduce a table
exhibiting in detail the complete calculation of such a table of
reduction for the standard wire employed by Dewar and
Fleming. We select for this purpose the following corrected
data, taken from their paper in the Phil. Mag., July 1895,
p. 100.
Thermometer in Melting Ice, B,° = 34059, t = Q°Q.
„ Steam at 760 mm., R / = 4'2034, £ = 100° C.
Liquid Oxygen, . R" = 0'9473, t 182°'5C.
From these data we deduce : —
Fundamental Interval, R' P° = 10975.
Fundamental Coefficient, (R'B, o )/100R° = 003533.
Fundamental Zero, y=l/c=283°'00.
In Liquid Oxygen,
p t=W6°'7, f=182°5, B = tpt = U°'2.
DifferenceCoefficient, d = I>/p(t) = U'2/5'lQ = 2'75.
DifferenceFormula, D = */?* = 2"75(*/100l)*/100.
To find the differenceformula in terms of pt, we have
similarly,
DifferenceCoefficient, d' = D/p(pt) = 14'2/5'84 = 2'43.
Pt DifferenceFormula, D' = *'7rt = 243(p*/100 1)^/100.
As a verification we may take the observation in solid C0 2
and ether, assuming Regnault's value t= — 78 0, 2 for the true
temperature.
DifferenceFormula (D) gives, tpt=2'75 x 139 = 3°82.
„ „ (D ; ) „ /'^ = 243xl49 = 3°62.
The observed value of ptis given as — 81 0, 9. Thus the
two formulae give, (D) f=s78°l, and (D') ^=78°'3, re
spectively.
The following Table shows the comparison of the formulae
for every ten degrees throughout the range. The first three
columns contain the whole work of the calculation for formula
(D') . The second column contains the values of D' calculated
by the aid of a small sliderule. These when added to the
values of pt in the first column, give the values of t shown in
the third column. The fourth column contains the correspond
ingvalues of the difference in t for 1° pt, obtained by differen
tiating the differenceformula. These are written down by
the method of differences. The fifth column contains the
difference t — t' between the values of / deduced by the two
formulae. The sixth contains the values of t by formula (D) ;
and the seventh is added for comparison with the table given
by Dickson (Phil. Mag., June 1898, p. 527).
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. Q
214 Prof. H. L. Callendar on Platinum. Thermometry.
Table II. — Table of Reduction for Dewar and Fleming's
Standard Platinum Thermometer.
pt (°C).
D'.
f(°0.).
i
dt/dpt.
tt'.
00
*(°C.)
+100
Dickson.
+ 9985
+ 100
+ 100
1024
+ 50
061
+4939
1000
008
+4931
+4947
+ o
•976
+003
 970
+ 020
 951
 10
+027
 973
•971
 20
+058
1942
•966
+005
1937
1918
 30
+095
2905
•961
+008
2897
2881
 40
+ 136
3864
•956
+011
3853
3839
 50
+ 182
4818
•951
+014
4804
4792
 60
+233
5767
•947
+017
5750
5742
 70
+289
6711
•942
+019
6692
6683
 80
+350
7650
•937
+022
7628
7625
 90
+415
8585
•932
+023
 8562
8561
100
+486
9514
•927
+025
9489
9492
110
+ 560
1044
•922
+026
1041
1042
120
+ 641
1136
•917
+026
1133
1134
130
+ 728
1227
•912
+025
1225
1226
140
+ 814
1319
•907
+024
1316
1317
150
+ 912
1409
•903
+ 022
1407
1408
160
+ 101
1499
•898
+019
1497
1498
170
+ 112
1588
•893
+016
1586
1588
180
+ 123
1677
•888
+011
1676
1678
190
+ 134
1766
•883
 +005
1765
1767
200
+ 146
1854
•878
! 002
I.
009
1854
1943
1855
1943
210
+ 158
1942
•874
220
+171
2029
•869
020
2031
2031
230
4 184
2116
•864
031
2119
2118
240
4198
2202
•859
 043
2206
2205
250
+213
2287
•855
1 058
2293
2291
260
+228
2372
•850
! 073
2379
2377
270
+243
2457
•845
090
2466
2463
280
+258
2542
•840
108
2553
2548
283
+264
2566
116
2578
2573
The above table affords a good illustration of the point
already mentioned, that the results obtained from the two
differ en cefornmlse (D) and (D') agree so closely over a limited
range, as in the present case, that it is often quite immaterial
which of the two is used for purposes of reduction. The
largest difference over the experimental range in the present
instance is only 0°"3, which is less than many of the errors of
observation, except at the fixed points and under the most
favourable conditions. In comparing the two formula the
following expression for the difference between them is
occasionally useful : —
DD / =^^'=^D(2^ + D100)/10,000+(^'l)D / .
Prof. H. L. Callendar on Platinum Thermometry. 215
It is generally sufficient to put D = D' on the righthand side
of this formula, so that if either is known the difference
between them may be determined with considerable accuracy.
It will be observed that the table of reduction given by
Dickson agrees very closely w r ith either of the difference
formulae. But, on the whole, most closely with (D). If
Dickson had calculated his formula from the same data it
would have given results identical with (D'). By giving
equal weight, however, to all the observations, without regard
to steadiness of temperature or probable accuracy, he is com
pelled, as in the previous instance, to admit an error of o, 35
in the fundamental interval itself, which is quite impossible.
Except at these points the probable error of his reduction is
not of vital importance ; on the contrary, the general agree
ment with (D) is so close that it is difficult to see on what
grounds he can regard the latter as being either incorrect or
inadequate.
For practical purposes a table of this kind is not convenient
owing to the continual necessity for interpolation. A graphic
chart in w T hich t is plotted directly against pt is objectionable,
because it does not admit of sufficient accuracy unless it is
plotted on an unwieldy scale. The difference curve avoids
this difficulty, and is much to be preferred for laboratory work.
But for occasional reduction it is so easy to calculate the
difference directly from the formula that it is not worth while
to take the trouble to plot a curve.
Reduction of Olszewski's Observations. — The observations
of Olszewski on the critical pressure and temperature and
boilingpoint of hydrogen, described in the Phil. Mag, for
July 1895, were made with a platinum thermometer of '001
inch w T ire wound on a mica frame in the usual manner. He
graduated this thermometer by direct comparison with a
constantvolume hydrogen thermometer at the lowest tem
peratures which he could obtain by means of liquid oxygen
boiling under diminished pressure. The lower temperatures,
observed with the thermometer immersed in temporarily
liquefied hydrogen, were deduced from the observed resist
ances by rectilinear extrapolation, assuming that the resistance
of the platinum thermometer continued to decrease, as the
temperature fell, at the same rate as over the lowest tempe
rature interval, —182*5 to —208*5, included in the range of
the comparison with the hydrogen thermometer. It is pos
sible that, at these low temperatures, the resistance of platinum
does not continue to follow the usual formula, but it may be
interesting to give a reduction of his observations by the
difference method for the sake of uniformity of expression.
Q2
216 Prof. H. L. Callendar on Platinum Thermometry.
We select for this purpose the following data : —
Thermometer in Melting Ice,
E/R°= 1000, *=0°C.
Thermometer in solid C0 2 at 760 mm.,
R/R° = 800, ^=782° C.
Thermometer in Liquid 2 at 760 mm.,
R/R° = 523, *=182°5 C.
From these we deduce the folio wing values of the coefficients : —
a=002515, b = 000,000,53, c = '002462, rf=213,
pi°=406°2.
As a verification we have the observation R/R° = *453, at
£=208°'5 C. This gives pt= 222% D=13°'7 ; which
agrees with the value given by the differenceformula calcu
lated from the three higher points.
The following Table owes the reduction of the observations
taken with this thermometer in partially liquid hydrogen.
Table III. — Reduction of Olszewski's Observations
in Boiling Hydrogen.
Pressure.
R/E°Obs.
pt.
D.
t (° C).
t Olszewski.
t'.
233°4
atmos.
20
•383
250°6
16°6
23°40
2345
10
•369
2563
173
2390
2397
2384
1
•359
2604
177
2427
2435
2420
The effect of this change in the method of reduction is to
make the temperature of the boilingpoint of hydrogen nearly
one degree higher than the value given by Olszewski. If we
employ instead the differenceformula in terms of pt, we should
find c' = 002472, pt° = 404°'5, d'=l'85. This formula leads
to the values given in the column headed t' , which are a little
higher.
The value found by Dewar for liquid hydrogen (Proc. R. S.
Dec. 16,1898) is much higher, namely t= — 238° # 8 at one
atmo, and — 239°*6* at l/30th atmo. The difference may
possibly be due to the superheating of the liquid, or, more
probably, to some singularity in the behaviour of his thermo
meter at this point (see below, p. 218) .
Observations of Holborn and Wien (Wied. Ann. lix. 1896).
— Holborn and Wien made a direct comparison between the
* Values calculated from observed resistances by formula (2).
gave
Dewar
Prof. H. L. Oallendar on Platinum Thermometry. 217
hvdrogen and platinum thermometers, adopting my method
of enclosing the spiral inside the bulb of the airthermometer.
The majority of their observations were taken while the tem
perature of the instrument was slowly rising. This method
of procedure is very simple, but it is open to the objection
that the mean temperature of the spiral is not necessarily
the same as that of the gas enclosed, especially when, as in
their apparatus, the spiral is asymmetrically situated in an
asymmetrical bulb. If we take their observations in melting
ice, in solid C0 2 , and in liquid air, which are probably in this
respect the most reliable, and calculate a differenceformula
in terms of pt, we shall find c' = *003621, d , = l'69. Calcu
lating the values of t' by this formula, we find that all the
rest of their observations make the temperature of the plati
num spiral on the average 1° higher than that of the gas.
This might be expected, as the temperature was not steady,
and the warmer gas would settle at the top of the bulb, the
spiral itself being also a source of heat.
If we take their own formula, and calculate the equivalent
differenceformula, we find c' = '003610, d'= 1*79. This agrees
fairly well with the values found above, as they appear also
to have attached greater weight to the observations in C0 2
and liquid air. But, if we take the formula calculated by
Dickson (Phil. Mag. Dec. 1897), who attaches equal weight
to all their observations, we find c' = '003527, d'=2'43. The
excessive difference in the values of the coefficients deduced
by this assumption is an index of the inconsistency of the
observations themselves*.
Behaviour of Pure Wire at Low Temperatures, — In the
case of ordinary platinum wire, with a coefficient c = '0035 or
less, the effect of the curvature at low temperatures of the
t, R, curve, as represented by the positive value of the dif
ferencecoefficient d, is to make the resistance diminish more
rapidly as the temperature falls, and tend to vanish at a point
nearer to the absolute zero than the fundamental zero of
the wire itself. When, however, the value of pt° is numeri
cally less than 273°, the effect of this curvature would be to
make the resistance vanish at some temperature higher than
the absolute zero. If, therefore, we may assume that the
resistance ought not to vanish before the absolute zero, we
should expect to find a singular point, or a change in sign of
the differencecoefficient, at low temperatures. If this were
the case, it would seriously invalidate the differenceformula
method of reduction, at least at low temperatures, and as
* Contrast the close agreement of Dickson's reduction in the case of
Fleming's observations.
218 Prof. H. L. Callendar on Platinum Thermometry.
applied to wires for which pt° was numerically less than
273°. When, therefore, I succeeded in obtaining in 1892 a
very pure specimen of wire, with the coefficient c = '00389,
pt° =257°, I quite expected to find it behave like iron and
tin, w T ith the opposite curvature to the impure platinum, and
a negative value for the coefficient d. On testing it at the
S.B.P. and also at the Ag.F.P. I found, on the contrary, that
it gave a value d=+l*50, and that its scale agreed very
closely with that of all the other platinum wires I had tested,
at least at temperatures above 0° 0. I sent a specimen to
Prof. Fleming shortly afterwards and he used it as the
mo thermometer P 9 " in his researches on the thermo
c
electric properties of metals at low temperatures. The test
of this wire is given by Fleming in the Phil. Mag. July
189 5, p. 101, from which the following details are extracted: —
c = 003885,jrt° = 257°4. C0 2 B.P., pt= 81°3.
O.B.P.,^=193°3.
Assuming *=182°5 at the O.B.P., we have ^=+2'10,
which gives £=— 78 c, 4 for the temperature of solid C0 2 .
The value of the differencecoefficient, so far from vanishing
or changing sign, appears to be actually greater at very low
temperatures. According to this formula, the resistance of
the wire tends to vanish at a temperature t° = — 240 o, 2, cor
responding to 2)t°= — 257°4. It seems not unlikely, however,
according to the observations of Do war, that the resistance,
instead of completely vanishing at this temperature, which is
close to the boilingpoint of hydrogen, ceases to diminish
rapidly just before reaching this point, and remains at a small
but nearly constant value, about 2 per cent, of its value at 0° C.
Application of the DifferenceFormula to the case of other
Metals. — The application of the differenceformula is not
limited to the case of platinum. It affords a very convenient
method of reduction of observations on the resistancevaria
tion of other metals. I employed it for this purpose in the
comparison of platinum and iron wires *, as a means of veri
fying the suitability of the parabolic formula for the expres
sion of variation of resistance with temperature. Thus, if the
symbol ft stands for the temperature by an iron wire thermo
meter, defined by formula (1), in exactly the same manner
as the platinum temperature, and if d and d' stand for the
differencecoefficients of platinum and iron respectively, as
suming that both wires are at the same temperature t, we
have clearly the relation
ftpt={dd f )Xp(t).
* PM1. Trans. A. 1887, p. 227.
Prof. H. L. Callendar on Platinum Thermometry. 219
As an illustration of the convenience of this method of re
duction a table is appended giving the values of the constants
at low temperatures for the specimens tested by Dewar and
Fleming. The data assumed in each case are (1) the value
of the fundamental coefficient c given in the first column, and
(2) the value of the temperature of the O.B.P. on the scale
of each particular metal, calculated from the observed re
sistance by formula (1), and given in the third column. The
value of the differencecoefficient d° for each metal as deduced
from the O.B.P. is found at once by the relation
^°=(_^1825)/516.
The sign of this coefficient indicates the direction of the cur
vature of the temperatureresistance curve, and its magnitude
is approximately proportional to the average relative curvature
over the experimental range.
The values of the coefficients a and b, given in the last two
columns, are readily calculated from those of c and d by
means of the relations already given (p. 199). These co
efficients refer to the equivalent resistance  formula ( 3 ),
and are useful for calculating the specific resistance at any
temperature.
In comparing the values of d°, given in this table, with
those deduced from observations at higher temperatures, it
will be noticed that they are in most cases algebraically
greater, the difference amounting to nearly 30 per cent, in
many cases between the values deduced from the O.B.P. and
the S.B.P. respectively. It is possible that this indicates a
general departure from the exact parabola requiring further
experiments for its elucidation. It would be unsafe, however,
to infer from the results of the present investigation that this
is always the case, because, owing to the construction of the
coils with silk and ebonite insulation, it was impossible to
test the wires directly in sulphur, and they could not be
annealed after winding at a higher temperature than 200°.
It is well known that annealing produces a marked effect on
the form of the curve and on the value of d*. It is also stated
in the paper that trouble was experienced from thermoelectric
disturbances, owing to the use of thick copper leads 4 mm.
in diameter. Such effects cannot be satisfactorily eliminated
except by the employment of a special method of compensa
* With reference to this point it is interesting to remark that Messrs.
Heycock and jNeville with one of their perfectly annealed pyrometers of
pure wire, for which c = '00387, e7= T497, found the value pt=— 80 o, 3,
t=— 78 0, 2C, for the C0 2 B.P. This would perhaps iudicate that the
larger values of d were due to imperfect annealing.
220 Prof. H. L. Callendar on Platinum Thermometry.
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Prof. H. L. Callendar on Platinum Thermometry. 221
tion, which will be described in a subsequent communication.
The general result of any residual thermal effects which may
be present is to produce a change in the apparent value of d,
since the thermoE.M.F. follows approximately a parabolic
formula. It is possible, for this reason, to obtain consistent
and accurate measurements of temperature with a platinum
thermometer in spite of large thermal effects, but the value of
d would be very considerably affected.
On the " Vanishing Temperature" — There appears to be a
very general consensus of opinion, based chiefly on the par
ticular series of experiments which are under discussion, that
the resistance of all pure metals ought to vanish, and does
tend to vanish at a temperature which is no other than the
absolute zero. If, hoAvever, there is any virtue in the para
bolic method of reduction, it is quite obvious, on reference to
the column headed " Vanishing Temperature " in the above
table, that the resistance "tends to vanish" in the case of
most of the common metals at a much higher temperature.
The vanishing temperature f is the value of t deduced from
the fundamental zero pt° in each case by means of the
differenceformula, employing the value of d° given in the
table. The most remarkable metals in this respect are pure
copper and iron, which tend to become perfect conductors at
a temperature of —223° approximately, a point which is now
well within the experimental range. These are followed at a
very short interval by aluminium, nickel, and magnesium.
In the case of copper and iron special experiments were made
at a temperature as low as —206° 0., at which point the
rate of decrease of resistance showed little, if any, sign of
diminution. The exact value of the vanishing temperature
in each case is necessarily somewhat uncertain owing to the
necessity of extrapolation, and also on account of possible
uncertainties in the data ; but there can be no doubt that the
conclusion derived from the formula represents, at least ap
proximately, a genuine physical fact. Whether or no the
resistance does actually vanish at some such temperature may
well be open to doubt. It would require very accurate ob
servations to determine such a point satisfactorily, as the ex
perimental difficulties are considerable in measuring so small
a resistance under such conditions. It is more probable that
there is a singular point on the curve, similar to that occur
ring in the case of iron at the critical temperature, at which
it ceases to be magnetic. It is also likely that the change
would not be sudden, but gradual, and that indications of the
approaching singularity would be obtained a few degrees
above the point in question. Below this point it is even pos
222 Messrs. Rosa and Smith on a Calorimetr
IG
sible that the resistance might not tend to vanish, but, as in
the case apparently of bismuth, might increase with further
fall of temperature. It has been suggested that at very low
temperatures all metals might become magnetic. It is very
probable that the change of electrical structure here indicated
would be accompanied by remarkable changes in the magnetic
properties. These are some of the points which experiment
will probably decide in the near future. The only experi
mental verification at present available is the observation of
Dewar in the case of platinum No. 3 when immersed in
boiling hydrogen at —240° C, that the resistance after at
taining a very low value apparently refused to diminish
further, in spite of a considerable lowering of the pressure.
It would be extremely interesting to repeat this observation
with specially constructed thermometers of copper or iron,
which ought to show the effect in a more striking manner
and at a higher temperature.
My thanks are due to Messrs. E. H. Griffiths, C. T. Hey
cock, and F. H. Neville, and to Prof. A. W. Porter and
Mr. N. Eumorfopoulos, for their kind assistance in revising
and correcting the proofs of this article.
X1Y. A Calorimetric Determination of Energy Dissipated in
Condensers. By Edward B. Rosa and Arthur W. Smith*.
IN a former paper {supra, p. 19} we gave the results of mea
surements by means of a wattmeter of the energy dissipated
in condensers when they were subjected to an alternating
electromotive force. The results were such that we desired
to confirm them by a totally independent method : and, in
addition, to measure the energy dissipated in some paraffined
paper condensers which showed so small a loss that with the
coils at our disposal the Resonance Method, employed success
fully on beeswax and rosin condensers, would not give
sufficiently accurate values. We therefore constructed a
special calorimeter for the purpose of measuring the total
quantity of heat produced in the condensers, which represents
the total energy dissipated.
Fig. 1 gives an external view of the calorimeter, and fig. 2
a vertical section. The calorimeter proper, A, is the inner of
three concentric boxes, and is 33 cm. long, 30 cm. deep, and
10 cm. in breadth. It has a copper lining, a, and a copper
jacket, b, and is protected by the two exterior boxes from
fluctuations of temperature without. The general principle
of the calorimeter is (1) to prevent any loss or gain of heat
* Communicated by the Authors.
Determination of Energy Dissipated in Condensers. 223
through its walls, and (2) to carry away and measure all heat
generated within by a stream of water To effect the first con
dition two concentric copper walls (the lining and the jacket)
are maintained as nearly as possible at the same temperature.
This, of course, will reduce the flow of heat through the inter
vening wooden wall to a minimum, and make the " cooling
correction " small, if not zero.
(i r>
1
To Eliminate the Cooling and Capacity Corrections.
In order to ascertain any difference of temperature between
the copper walls a and b, a differential airthermometer is
used. Each airchamber of this differential thermometer
consists of a copper pipe about 4 metres long and 4 millim.
internal diameter, one coiled about and soldered to the lining,
224 Messrs. Rosa and. Smith on a Calorimetric
and the other coiled ahont and soldered to the jacket
ne
end of each pipe is closed and the other connected to one end
of the Utube, G, shown on the outside of the calorimeter in
fig. 1. The Utube, which we call the gauge, contains kero
sene oil, and serves to indicate any difference of temperature
between the two copper walls. The zeromark is fixed after
Fig. 2.
E F
maintaining the whole calorimeter at a constant temperature
for some hours. In order to keep the gauge reading sensibly
zero, and thus keep the two copper walls very closely at the
same temperature, a coil of wire through which an electric
current of any desired strength can be passed is wound about
the jacket in the space B. And in order to make the regulation
more perfect a second coil is wound about the second box in
Determination of Energy Dissipated in Condensers. 225
the space C, so as to maintain the temperature of this space
nearly constant. The temperature of the chamber A is
usually kept a little higher than the external temperature, so
that no cooling is required ; and by varying the currents in
the two heatingcoils the temperature in B can be made to
follow that in A so closely that the gaugereadings are always
small, and their algebraic sum during any experiment zero.
This eliminates all correction for radiation. In rare cases
when the temperature of the room has risen considerably, we
have found it necessary to hang a wet cloth about the box to
prevent the temperature of C rising above that of B and A.
We intend to coil a small copper pipe in C so that a stream of
cool water may be sent through it, and then no difficulty will
be encountered in the hottest weather.
In addition to the gauge four thermometers (fig. 1) indicate
the temperatures of A, a, b, and C: that is, A' shows the tem
perature of the air in the calorimeter chamber A ; a! has its
bulb in a pocket of the lining a, and hence indicates the tem
perature of the wall a; V similarly extends down into a
pocket of the copper jacket b, and shows its temperature.
Finally, C 7 gives the temperature of the outer airspace C.
A' is an accurate thermometer reading from i0° to 25° C,
graduated to 0°*01 and read to o> 001 C. If A 7 shows the
temperature to be constant during the whole period of an
experiment, or the same for a considerable time near the end
of an experiment that it was at the beginning, then there will
be no correction for heat absorbed or given up by the appa
ratus. With both the " cooling correction " and the capacity
correction eliminated, it remains to carry away and measure
the entire heat generated by a condenser in A, or by any other
source of heat within the calorimeter.
2. Carrying away and Measuring the, Heat.
In order to carry away the heat generated a stream of
water, which enters at 1 (figs. 1 and 3), is made to flow
through a coiled copper pipe (fig. 3), where it absorbs heat,
and then leaves the calorimeter at 0. In order to increase its
absorbing capacity the pipe is soldered to a sheet of copper,
L L, both pipe and copper being painted black. Three such
sheets, each with 4 metres of pipe attached, are joined
together and placed side by side in the chamber A, the con
densers being slipped in between them. The rate of absorption
of heat depends upon the difference of temperature between
the absorbers and the air surrounding them. If a large
226
Messrs. Rosa and Smith on a Calorimetric
amount of heat is to be brought away, the water is made to
enter at a low temperature and to flow rapidly through the
absorbers. If a smaller quantity of heat is to be absorbed
and carried away, the entering water will be warmer, and its
gain in temperature correspondingly less. By varying the
temperature of the water and its rate of flow, the rate of
absorption can be varied between wide limits, and kept very
Fig. 3.
nicely at any desired point. In practice the thermometer A'
is the guide in regulating the temperature of the entering
water. If the temperature of A begins to rise (A 7 , as already
stated, can be read to onethousandth of a degree), the entering
water is slightly cooled; if to fall, it is slightly warmed ; the
rate of flow of water, after being once adjusted for a given
experiment, is maintained constant.
In order to measure the quantity of heat thus carried
away, the thermometers E and F are inserted in two small
reservoirs, M and N, which stand in the wooden wall of the
calorimeter between the two copper surfaces. The thermo
meter E indicates the temperature of the water just as it
enters the chamber A, and the thermometer F gives its tem
perature as it leaves. The difference of temperature multiplied
Determination of Energy Dissipated in Condensers. 227
by the mass of water per second gives the rate of absorption
and removal of heat. The thermometers are accurately gra
duated and read to hundredths of a degree. The gain in
temperature is several degrees, and may be ten or twenty
degrees by increasing the quantity of heat generated or
reducing the rate of flow of water. Hence the accuracy of
the determination of the quantity of heat absorbed is sufficient
for most purposes. The chief error is ordinarily due to
changes in the temperature of the apparatus itself and its
contents. By running the experiment several hours, however,
and keeping it as nearly as possible at a constant temperature,
this uncertainty is greatly reduced and the error made
negligibly small.
The water flows into the calorimeter from a reservoir about
a metre above, this height furnishing the necessary pressure.
The temperature of the entering water is regulated by adding
warm or cold water to the reservoir, and the rate of flow of
the water is regulated by an adjustable valve. The water is
collected in a litre flask, the time of each litre being recorded.
3. Test of the Calorimeter.
Table I. shows the result of one of the tests made upon the
calorimeter. A current of electricity passed through a coil
of wire within the chamber A, the electromotive force being
measured by a carefully calibrated Weston voltmeter, and
the current by a Kelvin balance. The experiment continued
for a little more than four hours, while nine litres of water
passed through the calorimeter. The rate of absorption of
heat was nearly, but not quite, constant, the temperature as
indicated by A! having varied slightly. The final tempe
rature was practically the same as that at the beginning, being
slightly higher if anything. The average for the nine litres
is 12*37 watts absorbed and carried away by the water, while
the electrical measurements give 12*34 watts. By continuing
the experiment longer and introducing greater refinements in
the measurement of the current and electromotive force, a
greater degree of accuracy could undoubtedly be attained.
But this and other tests showed clearly that for our present
purposes the calorimeter was abundantly accurate, and we
proceeded to put some condensers into it and measure the heat
evolved.
228
Messrs. Rosa and Smith on a Calorimetric
Table I.
Test of the Calorimeter.
(ft)
(<0
Period in
seconds
Time.
for each
lOOOgrm.
h. m. s.
of water.
1 53 00
2 19 45
1605
2 46 50
1625
3 13 55
1625
3 41 15
1640
4 09 10
1675
4 37 10
1680
5 04 10
1620
5 32 15
1685
6 01 00
1725
i
(d)
Average
tempera
ture of
the
ingoing
water.
(«) (/)
Average ' Increase
tempera in tempe
1566
1570
1574
1570
1572
15 57
1564
1555
1557
I ture of
 the out
i i going
water.
rat lire of
I each
lOOOgrm.
I of water.
\{e){d).
(A)
iff)
Total
heat
measured
(small
calories).
(/)X1000. (g)^(c)
Small
calories
per
second.
2039
2044
2051
2053
2064
2056
2058
2050
2056
o
473
474
477
483
492
499
494
495
499
4730
4740
4770
4830
4920
4990
4940
4950
4990
2947
2917
2935
2945
2937
2970
3050
2950
2893
(0
Equivalent
watts =
calories X J.
(70x41972.
1237
1224
1232
1236
1233
1247
1279
1233
1214
1237
Electromotive force =20*0 volts.
Current=0617 ampere.
Watts (from electrical measurements)
200x0617= 1234.
4. The Experiments.
Table II. (p. 230) give? the results of six experiments with
the same beeswax and rosin condensers which were employed
in onr work by the resonance method. In each experiment a
preliminary run, not included in the table, allows the condensers
and calorimeter to come to a constant temperature. Column
(a) gives the numbers of the condensers in each case, they all
being joined in parallel to the same electromotive force.
Column (b) gives the time of the beginning of each litre of
water; column (c) the duration of each litre or 1000 grammes
of water; column (d) the average temperature of the ingoing
water as found from readings of the thermometer E, taken
regularly every five minutes, and column (e) the same for the
outgoing wafer; (/) then shows the increase of temperature.
Column (g) gives the number of calories of heat carried away
by each 1000 grammes of water, and column (Ji), which is the
number in (g) divided by the corresponding number of seconds
recorded in (c), is the rate of absorption of heat. Column (i)
gives the number of watts to which this is equivalent, taking
Determination of Energy dissipated in Condensers. .229
J, the mechanical equivalent of heat, to be 41,972,000 ergs.
This is the value derived from Rowland's and Griffiths's work,
assuming the specific heat of water at an average temperature
of 20° C. to be unity. Column (j) gives the frequency.
Sometimes this was estimated from the average frequency of
the dynamo supplying the lines of the Middle town lighting
circuits at the time ; and in other cases it was determined by
measuring the speed of a small synchronous motor. The
electromotive force (k) was measured with an electrometer, the
current (I) with a Siemens dynamometer. Column (??) gives
the values of cos cf> of the expression power = EI cos <£. Care
was taken in every instance to avoid the presence of upper
harmonics, in some cases using a resonancecoil to quench the
harmonics as well as increase the voltage on the condenser.
Column (o) gives the per cent, loss, 1007T cot </>, and column
(p) the net efficiency. This relative loss, tt cot <$>, has been
proved* to be the ratio of w to W, where w is the energy
dissipated per halfperiod, and W is the energy stored in the
condenser at each charge. 1— 7rcot<£, the net efficiency, is
therefore
Energy stored — Energy lost
Energy stored
5. Beeswax and Rosin Condensers.
The first experiment, with condensers Nos. 3, 4, and 7,
showed a net efficiency of 93*39 per cent., or a loss of 6*61
per cent. The temperature of the condensers was not deter
mined ; but from the fact that the dielectric was softened and
the quantity of heat generated was more than in any suc
ceeding experiment, we feel sure that it was considerably
above 40° 0. Six condensers were then placed in the calori
meter, joined in three pairs. Nos. 1 and 5 gave no sound
when joined to an alternating E.M.F. of 1000 or more volts,
and we called it the " best pair/' In Nos. 3 and 4 vibrations
were distinctly felt when the fingers were placed in contact
with them, while the condensers gave a clear musical note
and on the higher voltages a hissing sound ; this we called
the " poorest pair." Nos. 8 and 9 were intermediate.
These six condensers were first of all joined in parallel and
connected to a lowfrequency circuit of 1520 volts and 26
periods per second. Care was taken to exclude upper har
monics. The experiment continued over three hours after
the temperature of calorimeter and condensers had become
* See our paper, Phil. Mag. Jan. 1893.
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. R
230
Messrs. Rosa and Smith on a Calorimetric
Table II. — Measurements of Energy dissipated
(«)
P)
M
(d)
(e)
(/)
Increase
(9)
(A)
No. of
the
con
Period in
Average
Average
in tempe
Total heat
Small
calories
per second.
seconds for
tempera
tempera
rature of
measured
Time.
each
1000 grams
ture of the
ingoing
ture of the
outgoing
each
1000 grams
(small
calories).
LIO.LLOC1 .
of water.
water.
water.
of water.
(e)(d).
(/)X1000.
(<?)(*).
(1)
h m 8
9 36 00
o
6
o
3, 4,
10 02 00
1560
1576
2084
508
5080
3256
&7.
10 25 00
1380
1576
2072
496
4960
3595
10 51 30
1590
1587
2014
427
4270
2686
(2)
6 34 35
1&5,
3&4,
8&9.
7 12 30
2275
1778
2207
429
4290
1886
7 49 15
2205
1757
2208
451
4510
2045
8 26 30
2235
1694
22 08
514
5140
2300
9 05 10
2320
1659
2197
538
5380
2319
9 41 40
2190
1623
2192
569
5690
2598
(3)
2 56 30
1&5,
3&4 S
8&9.
3 27 10
1840
1762
2160
398
3980
2163
3 55 45
1715
1771
2154
383
3830
2233
4 24 00
1695
1761
2148
387
3870
2283
4 52 15
1695
1766
2145
379
3790
2236
5 21 20
1745
1758
2143
385
3850
2206
(4)
7 10 40
7 28 13
1653
1765
2158
393
3930
2378
1 & 5.
7 56 40
1707
1771
2162
391
3910
2291
8 25 47
1747
1770
2166
396
3960
2267
8 55 50
1803
1768
2169
401
4010
2228
9 26 20
1830
1763
2166
403
4030
2202
(5).
2 38 30
3 11 30
1980
2064
2265
201
2010
1015
1 &5.
3 44 20
1970
20 90
2262
202
2020
1025
4 17 20
1980
2066
2258
202
2020
1020
4 50 20
1980
20 68
2264
196
1960
0990
;
(6)
8 37 20
1
3&4.
9 15 20
2280
2055
2227
172
1720
0754
9 54 15
2335
2059
2228
169
1690
0724
Determination of Energy Dissipated in Condensers. 231
in Beeswax and Rosin Condensers.
(*")
U)
(*)
(0
(m)
(n)
(*)
(P)
Equivalent
watts
=average
calories X J.
Fre
quency.
Electro
motive
force
(volts).
Current
(amperes).
Apparent
watts.
COS0.
Per cent
loss =
7TCOt(pX 100.
Efficiency =
(lTTCOt^)XlOO.
(A) X 41972.
E.
I.
ExI.
(••)■*(»).
(»)X«rXl00.
100 (o).
1334
140
868
•730
634
•0210
661
9339
(30°)
936
26
1520
•400
608
0154
484
9516
(30°)
933
120
650
•897
583
•0160
508
9494
(40°)
954
140
805
•445
358
•0266
837
9163
(30°)
425
137
630
•333
210
•0202
635
9365
(30°)
310
140
605
•333
202
•0154
482
9518
R2
232 Messrs. Rosa and Smith on a Calorimetric
constant by a preliminary run of several hours. The tem
perature of the calorimeter as indicated by the thermometer
A' rose gradually for an hour, and hence the heat absorbed
was less than the average. During the last hour the tempe
rature was reduced by quickening the rate of flow and cooling
the entering water, so that the temperature was substantially
the same at the end as at the beginning. The per cent, of
loss is 4*84 at an average temperature of the condensers of
30° C. The voltage employed on this lowfrequency test was
much higher than for any other experiment, and yet there
was no evidence of brushdischarge or appreciable leakage
current.
In the third, experiment the same condensers were subjected
to a highfrequency electromotive force at 30° C, and the loss
found to be 5*06 per cent., that is, slightly greater than before.
Hence for a given voltage the energy dissipated, per period
would be slightly greater, and the energy dissipated per second
more than five times as much as for the low frequency.
In the fourth experiment only the " best pair " of con
densers was used, and with a slightly higher voltage the
temperature of the condensers rose to 40° C. Here the los3
was found to be 8*37 per cent., nearly as much as the maximum
value found by the resonance method.
The fifth experiment, with the " best pair," was made
some days later at 30° C, and the percentage loss came out
6*35 per cent., that is greater than the average of the six. This
was unexpected, as well as the last result, which showed a loss
for condensers 3 and 4, the " poorest pair," of 4*82 per cent.,
which was less than the average. These results were then
confirmed by an independent method, showing conclusively
that the socalled " poorest pair " had the smallest loss ; not,
of course, because it emitted a distinct sound and hissed on
high voltages, but in spite of that. The chief loss is doubtless
due to some cause quite independent of the singing and
hissing, and happens to be smaller where it would naturally
be expected to be larger.
Thus we have confirmed by these calorimetric measure
ments the large values of the losses which we found by the
resonance method in beeswax and rosin condensers, and also
the existence of a wellmarked maximum as the temperature
rises, beyond which the loss decreases considerably. It is an
interesting fact that the residual charges of these condensers
are very large, that they increase with the temperature up to
40° C, and then decrease as the temperature is carried
higher. That is, the maximum point for the residual charge is
the same as for the energy loss.
Determination of Energy dissipated in Condensers. 233
6. Paraffined Paper Condensers.
The second lot of condensers used were commercial paraffined
paper condensers made by the Stanley Electric Co. A finished
condenser is a solid, slab about 25 x 30 cm. and 2 cm. thick,
thus having a volume of 1500 c. c, and is enclosed in a tight
tin case, the leadwires coming out through ebonite bushings.
Nos. 1 to 4 of our condensers have a capacity of about
1*7 microfarad each ; Nos. 5 to 10, which were purchased
about a year later, have a capacity of about 3*2 microfarads
each. The condensers of the second lot are made of paper
about 0038 cm. thick, two sheets being placed together in
each stratum. This we learned by dissecting some which we
had broken down. It ought to be stated, however, that
while the condensers are guaranteed by the makers to stand
500 volts alternating electromotive force, we have repeatedly
subjected them to 1000 to 2000 volts, and in some cases for
several hours at a time. Nos. 9 and 10 were upon one
occasion maintained at 2250 (effective) volts, at a frequency
of 130, for over an hour, and showed no signs of being over
taxed. We have, however, broken several at voltages between
1000 and 2000. The paper of the first lot of condensers is
thicker, but as we have never broken one of this lot we
cannot state its thickness. From the fact that the capacity
of each of these is about 60 per cent, as great as that of the
others, while their volumes are substantially the same, we
conclude that the thickness of the paper is about "0048 cm.,
supposing there are, as in the other, two sheets together in
every stratum of the condensers.
In Table III. are given the results of seven separate expe
riments with Stanley condensers, which were made at intervals
during the past three months. The frequency in every case
except experiment 4 was estimated from the average frequency
of the two dynamos of the Middletown lighting circuits.
Experiments 1 and 3 were made when the faster dynamo was
supplying the lines, the others were with the slower dynamo.
All but No. 4, however, were with a relatively high frequency.
No. 4 was made using a twopole rotary transformer, sup
plying it with direct current, and running it at a speed of
1600 per minute. The percentage losses (o) vary more among
the different condensers at the same frequency than one would
expect.
The percentage loss at the frequency 28 (Experiment 4) is
'78 per cent., whereas at a frequency five times as great it is
(Experiment 3) 1*00 per cent. At 120 it is, as would be ex
pected, nearly as great as at 140; that is, it is *96 per cent.
234
Messrs. Rosa and Smith on a Calorimetric
Table III. — Measurements of Energ) T
(a)
(6)
w
(d)
w
(/)
Increase
(o)
(h)
No. of
the
Period in
Average
Average
in tempe
Total heat
Small
calories
per second
seconds for
tempera
tempera
rature of
measured
Time.
each
ture of the
ture of the
each
(small
con
denser.
1000 grams
ingoing
outgoing
1000 grams
calories).
of water.
water.
water.
of water.
(•)<<&
(/)X1000.
(t)H*y>
(1)
h m s
7 24 00
o
O
o
1.
7 59 50
2150
1971
2412
441
4410
2057
8 36 30
2200
1981
2411
430
4300
1955
(2)
9 13 30
2220
1982
2420
438
4380
1973
12 33 25
1 04 00
1835
1806
2330
524
5240
2856
3&4,
1 36 00
1920
1804
2343
5 39
5390
2808
2 07 40
1900
1793
2340
547
5470
2879
2 40 00
1910
1798
2347
549
5490
2830
(3)
7 17 20
7 46 20
1740
1671
2209
538
5380
3092
3&4.
8 15 50
1770
1659
2212
553
5530
3124
8 45 23
1773
1633
2211
578
5780
3260
9 14 43
1760
16 55
2206
551
5510
3131
(4)
12 21 12
3&4.
1 21 25
3613*
18 45
2246
401
2807*
0777
2 37 30
4565
1867
2247
380
3800
0832
3 47 35
4205
1880
2246
366
3660
0870
(5)
12 24 20
6.
12 55 10
1850
1944
2325
378
3786
2140
1 26 45
1895
1955
2320
365
3650
1926
1 59 30
1965
1948
2320
372
3720
1893
(6)
6 05 08
10.
6 41 30
2182
17 82
2248
466
4660
2136
6 54 30
780 1
1810
2280
470
1739 1
2231
(7)
2 17 55
2 50 00
1925
1796
2235
4 39
4390
2281
10.
3 21 30
1890
1805
2239
434
4340
2296
3 53 28
1918
1798
2244
446
4460
2325
4 25 00
1952
1792
2244
452
4520
2315
* The amount of water for this period was 700 grams.
t „ .. „ 370 „
Determination of Energy Dissipated in Condensers. 235
dissipated in Stanley Paraffin Condensers.
(0
Equivalent
watts
= average
calories x J
(h) x 41972.
836
1193
J 323
3 47
834
917
967
(?)
(A)
(0
O)
(»)
Fre
quency.
Electro
motive
force
in volts.
Currrent
in amperes.
Apparent
watts.
COS0
E.
I.
EI.
(i) + (m).
140
1133
160
1813
•0046
120
1264
31
3918
•0030
140
1194
35
4179
•0032
28
1659
•837
1389
■0025
120
778
234
1822
•0046
120
1294
30
3882
•0024
120
1294
30
3882
•0025
(p)
Per cent.
loss =
■k cot <p X 100.
(P)
Efficiency =
(l7TCOt0)XlOO.
0)xttx1oo. loo (o)
145
•96
100
•78
144
•74
•78
9855
9904
99 00
9922
9856
9926
9922
236 Prof. K. Pearson on certain Properties
(Experiment 2). Condenser No. 1, of the same lot, shows a
loss of 1*45 per cent, at frequency 140, which is 45 per cent,
greater loss than Nos. 3 and 4 give. Of the second lot, No. 6
gives a large loss (Experiment 5), and other experiments
which one of us has made by other methods show that all the
other condensers of this lot have losses nearly the same as No. 6,
excepting No. 10, which gives the smallest loss of any, *74 per
cent, in one case and '78 per cent, in another (Experiments 6
and 7). Condenser No. 2 shows by other methods the same
loss as 3 and 4. Hence we have the following singular
results : — All of the first lot except one have a loss of 10 per
cent, on high frequencies, and the exceptional condenser has a
loss of 1*45 per cent. All the condensers of the second lot have
substantially the same losses, about 1*5 per cent., and the excep
tional one is scarcely more than one half as much as the others;
the exceptional one of the first lot having the same loss as all but
one of the second. There is no possibility of a confusion of
numbers, for they were plainly stamped when purchased, and
the capacities of the first and second lots are very different, as
already stated. Our experiments do not indicate the reason
for these large differences ; but the existence of such differ
ences is fully confirmed by measurements made by wholly
independent methods, and which will shortly be published.
Wesleyan University,
Middletown, Conn., Sept. 1, 1898.
XV. On certain Properties of the Hyper geometrical Series,
and on the fitting of such Series to Observation Polygons in
the Theory of Chance. By Kabl Pearson, F.R.S.,
University College, London*.
1. TN a paper entitled " Mathematical Contributions to
JL the Theory of Evolution : Part II. Skew Variation
in Homogeneous Material " t, I have pointed out that the
following series, of which the skewbinomial is a special case
(w=oo),
pn(pnl) (pn — 2) .... (pn— r + 1)
n(n — l)(n — 2) (n — r+1)
( x ! r 9 n , r(rl) gn(qnl)
V pn — r + 1 1.2 (pn — r + l)(pn—r + 2)
r(rl)(r2) qn{qnl){qn2) ^
1.2.3 {pn — r + l)(pn — r + 2)(pn — r + 3)
* Communicated bv the Author.
+ Phil. Trans, vol. clxxxvi. p. 360 (1895),
of the Hyper geometrical Series. 237
is especially adapted for fitting various types of frequency
distribution. The relative magnitude of r and n is, indeed,
often a very good test of the " interdependence of contribu
tory causes."
If we put
a=—r,/3=—qn,y=pn — r+l ... (2)
and denote by F(a, /3, y, x) the general hypergeometrical
series
1+ ^ + ^±l»+i)^ +&c . . . . (3)
1.7 1.2.7(7+1)
we see that our series is a hypergeometrical series of the type
F(a, j3, 7, 1), or, as we shall denote it F 1 (a, ^7), multiplied
by a factor, which we may write A .
If the successive terms of a hypergeometrical series be
plotted up as ordinates at intervals c, and the tops of these
ordinates be joined, we obtain a great variety of polygons,
which approximate to the interesting series of generalized
probabilitycurves with which I have already dealt. The
advantage of the hypergeometrical polygons over the curves
consists in the knowledge as to the nature of the chance
distribution indicated by the discovery of the actual values
of p, q, n, and r. The curves, however, possess continuity
and are easier of calculation. Clearly a knowledge of a, /3,
y, since 7 — a— /3 — l = n, gives n, and hence g, r, and p.
We shall find it convenient to write
m 1 = a + ^, m 2 = a/3. . . . . . (4)
It is not, however, only in the question of distribution of
frequency that hypergeometrical series may be of service ;
it seems extremely probable that the three constants #, /3, 7
of Fj (a, (3, 7) may be of service in indicating close empirical
approximations to physical laws, owing to the great variety
of forms that the hypergeometrical polygon can take.
Before we proceed to the fitting of hypergeometrical
polygons to given data, we require to demonstrate one or two
general propositions with regard to such figures.
2. On the moments of Fi(a, /3, 7). — Let A= the area
of the polygon, thus if the ordinates are plotted at distance
c, we have A = cxF 1 . Let fx s K be the sth. moment of
the polygon about its centroidvertical, the elements of area of
the polygon being concentrated along the ordinates. Let v 8 'k.
be the sth moment of the ordinates about a vertical parallel
and at a distance c from the first ordinate, i.e.
238 Prof. K. Pearson on certain Properties
Now let a new series of functions xoy %\> Xii & c  De
formed, so that
andletx =F(a, /3, y, a).
Then we have
v s = c s x(xJXo) x= r
fi s can then be found from v s v 3 _ v v s _ 2 , &c. by the formulas
given on p. 77 of my memoir (Phil. Trans, vol. clxxxv.).
Thus the determination of the successive moments of the
hypergeometrical series F t is thrown back on the discovery
of the %'s from the value F.
3. To find the successive %'s. — The hypergeometrical series
is known to satisfy the differential equation
x d / d¥\ . ,„ . x dF ^ .
(see Forsyth, ' Differential Equations/ p. 185).
But dF
Hence, substituting and rearranging, we have
(l^){% 2 +(m 1 2)xi+(m 2 m 1 + l)%o}+wXi( n  , ^)Xo = 0. (6)
Put #=1, we have
(%i)i=^ (Xo)i (7)
or rc + m 2 /ON
This is the distance of the centroid vertical of the hyper
geometrical series F 1 from the vertical about which the
vmoments are taken.
Multiply (6) by x and differentiate, we find
(l^){^3 + (m 1 2)x 2 +(w2^i + l)%o}^{X2+( m i2)x l
+ (m 2 — m x + l)x<>} + n Xi  ( n + m 2>Xi = 0,
or
(1^){X3+ ( m l~l)%2+(^2l)%l+(^2~Wl 1 + l)Xo}
+ ( 7l  1 )X3( n + m i + w 22)x 1 (m 2 w 1 + l)%o = . (9)
• of the Hyper geometrical Series. 239
Put a?=I, we have
( 7l  1 )(%2) 1 = (n + w 1 + m 2 2)x 1 + (w 2 ~m 1 4l)(%o)i
. . n 2 + n(5m 2 — 1) + m 2 2 + m x m 2 — 2m 2 /1A .
= (Xo)i , • (10)
by aid of (7).
l^ us ri 2 + n(3m 2 — l)+m 2 2 + 77? 1 m 2 2m2 /1f , N
v 2 = ^ t fr — . • (11)
n(n — 1) v 7
and o
^ 2 = v 2 v 1 J
_^ 2 m 2 (n 2 + m 1 n + m 2 )
or, we may write
c W?(n + «)(n + / 3) (1)
:* n 2 (n — 1) v y
Multiplying (9) by # and differentiating again, we find
{lx) {x± + m^ +{m 1 + m 2 2)X 2 +(2m 2 — m 1 )xi + (m 2 —m 1 + l)xo}
+ (n— 2)^— (71 + 2772! + w 2 3)x2 — (2w 2 — m 1 )%i~(m 2 —m 1 + l)xo = 0. (14)
Putting a?=l, we have
(n2)(x 3 )i=(n + 2m 1 + m 2 3)( %2 ) 1 + (2772277?!) (^Oi
+ (2722772! + 1) (% )l,
or, by aid of (7) and (10),
v 3 = c 3 {?2 3 + n 2 (7m 2 — 3) + n (6tt2 2 2 + 67Bi772 2 — 15m 2 + 2) + m 2 3
+ %m Y m 2 2 + 2m l 2 m 2 — 7m 2 2 — Gm^g + 6m 2 }4w(?2 — 1) (72 — 2) . (15)
Hence, since fi z = v 3 — Sv^— Vi 3 , we have after some reduc
tions
_ c 3 xj3(n + *)(n + {3Xn + 2a)(n + 2l3)
P» » 8 (»rl)(n2) ' ' ' U j
Differentiating (14) after multiplication by a?, we find
(l^)X5+K + l)X4+(2m l + m22)x3+(3m 2 2) %2
+ (3m 2 — 2m 1 + l)xi+(w 2 — m 1 + l)xo} + (w— 3)^4
— (71 + 3772! + m 2 — 3)X3~(3w 2  2)% 2 —(3772 2 — 2772! + 1)^!
 (7722772! + 1) %0 = (17)
Putting * =3=1 j we have
(n  3) ( X4 )x = (n + 37?ii + m,  3) Gfc) i + (3m 3  2) ( %2 ) j
+ (3m, — 2m 1 + l)(xi)i+ (wiamx+lXxoJi.
240 Prof. K. Pearson on certain Properties
Hence, by the use of (7), (10), and (15), we deduce
v^ = c 4 {n 4 + n 3 (15m 2 — 6) + rc 2 (25m 2 2 + 25m 1 m 2 — 65m 2 4 11)
+ n(10w? 2 3 +30m 1 ?72 2 2 + 2()m 1 2 ?^ 2 — 75m 2 2 65m 1 w 2 + 80m 2 6)
4 m 2 + 6w 1 m 2 3 + llwi 1 2 ?w 2 2 + 6m l 3 m 2 — 16m 2 3 — 42wiW 2 2
— 24m 1 2 /?? 2 f S6m ] m 2 + 49w 2 2 — 24m 2 }
in(nl)(n2)(n3). . . (18)
But 2
thus we find
c 4 m 2 (n 2 + m 1 nf w? 2 ) . , ,,., „ .,
* = n'(»l)(n2)(nV X {**+«>(*»» + *".*U
+ rc 2 (3mim 2 + 6mi 2 + Qm 2 ) + n(3m 2 2 + lSm^m^ + 18m 2 2 }. (19)
Now a = — r, /3 =—qn ;
m x r r\ Wo .
n \ ' n> n
Substitute these values in (19), and make n infinite. The
hypergeometrical series now becomes the binomial (p + q) r i
and we have
/*t = c\l + 3(r2)pq),
a result already deduced (Phil. Trans, vol. clxxxvi. p. 317).
This serves to confirm (19).
Dividing equation (19) by (1 — #), and putting = 1, we
find, by remembering that
r r s)„=(^)„r{S}„" te " , " :
("  4) to) l => + 4m! + m 2  2) ( %4 ) x + (2mj + 4m 2  4) ( %3 ) x
+ (6m 2 2m!l)( %2 ) 1+ (4m 2 3m 1 +2)( %1 ) 1
+ (/7? 2 mi + l)(xo)i . . •
Whence
c
v 5 — —, tyt 577 577 rr { ft 5 + n 4 (3 lm 2 — 1 0)
n(n— l)(w — 2)(n— 3)(n— 4) l v
+ n 3 (90m 2 2 + 90m!m 2 — 220m 2 + 35)
+ n 2 (65m 2 3 + 195m 1 m 2 2 + 130mi 2 m 2  485m 2 2
420m 1 m 2 + 535m 2 50)
*•»
t 2 — uitJ//o.l7l 2 ""Oil
+ 800m 2 2 490m 2 + 24) + m 2 5 + 10miW 2 4
f 35m l 2 m 2 * + 50m l 3 m 2 2 + 24ra 1 4 m 2 — 160m 1 m 2 3
 256mi 2 m 2 2 i20m 1 3 m 2 + 240™^ + 490/*!m 2 2
240m!m 2  30m 2 4 + 213tw 2 3  380m 3 2 + 120m 2 }, (20)
+ n(15m 2 4 + 90??*im 2 3 +165m 1 2 m 2 2 + 90mi 3 m 2
645m.m 2 2 — 370m 1 2 m 2 f 600m 1 m 2
of the Hyper geometrical Series.
241
Pi = v 6 — 5 Vi/^4 — 10 vfps — 10 j/ x 3 ^2 — V
_ c 5 m 2 (n* f ti^ 4 77? 2 ) (t? 2 + 2m 1 n + 4m 2 )
7i 5 (nl)(>i2)(n3)(7i4)
X {n 4 + ?i 3 (10?7i 2 +12m 1 + 5) +w 2 (10m 1 m 2 + 127/i l 2 )
+ n(10m 2 2 + 24m 1 m 2 ) + 24m 2 2 }. . .
To determine the value of n, m u and m 2 , let us write
A=A t 3 2 /^ 3 , A=/*V/*s 8 i ^3=^5/(^3^2) ;
and to render the elimination easier, let us put
e = w 2 f nmx + m 2 , '
m 2 e = 2 2 , V (22)
e\m 2 — z x .
Then, from (13) and (14),
(21)
A(Mt ( ^^
(23)
From (19) and (14),
From (21), (13), and (13),
ft = (^p^ {^ + 5 ^+^(10n24) + 12(* 1 »n«* l )}. (25)
These are linear in 2 2 ; collecting all the terms of ^ 2 on the
left, we can rewrite (23) to (25),
*2#1
(n2V
711
— ^4
7l 4 + 4^ 1 2 7l 2 < 2r 1 ),
(26)
zA/3 2 {n ^n3) _ (3h _ 6) J = »*+!!■ + e^^), ( 27 )
z 2 {g 3 (n ""^" 4) (10n24) 1 = n« + 5n'+12fa»n« g ). (28)
Multiply (26) by 3, and (27) by 2, and subtract the results
from (28) :
{
'•Jfl
(n3)(n4) „„ (n2) 2
1
3/8,
!L_2L_(i0n_24)l=2» 4 + 5n 3 , . (29)
^k (W  8) _ ( r 4)  2 /3^ "^r 3) (4n12)Un 4 + 3n3.
(30)
242 Prof. K. Pearson on certain Properties
The last equation will divide by n — 3, or
{^W^^nK
(31)
Substitute this value of n 3 on the righthand side of (29),
and we have
A ( w 3K,4) _ 3A (^2) ! _ (1()n _ 24)
or
w 2
Divide out by and we have
J n — 1
&(n4)2A(2/i5) + 3ft(n2) + 2(nl)=0,
or
^(A4 y 5 2 + 3/3 1 + 2)=4y5 3 10A + 6/3 1 + 2.
Thus
4^310^ + 6^ + 2
n  A4ft + 3A + 2 (32)
n being now known (31) gives us
n*(n — 1) ,„„v
* 2 4(rcl) + 2/3 2 (^2)/3 3 (rc4) ; " ' K ° 0)
z 2 and n being known, we have bv (23)
^kW^s^ • • • ^
Then ra 2 and e are the roots of the quadratic
? 2 2i? + *2=0 (35)
m 2 and e being known, we have
e — m 2 — n 2
™i = " (36)
Next, « and /3 are roots of
f2_ mi ^ +m2==0 , (37)
and y=n + « + /3+l. . ..... (38)
Lastly, from (2)
r=«, q=fi/n i and ya ?"*" 1 , , (39)
of the Hypergeometrical Series. 243
Thus, all the constants p, g, r, and n of the series (1) are
determined.
The base unit c is given by (12), or
e=w Je&=R (40)
To obtain the successive ordinates of the hypergeometrical
frequencypolygon we must, if A be the total number of
observations, take the successive terms of
1 + r qn + T(r " 1]  9<gnl) &c>
pn — r+1 1.2 (pn — r+1) [pn— r + 2)
multiplied by A, or
Apn(pn — l)(pn — 2) . . . (pn— r + 1)
c n(n — l)(n — 2) . . . (n — r + 1)
The position of the first ordinate is at a distance d^v x —c
from the mean (or centroid vertical) of the series, i. e.
d=cm 2 /n (41)
Thus the solution is fully determined. Its possibility
depends on positive and real values for n and r, and for p
and q.
As an illustration I take the following data provided for
me several years ago by members of my class on the theory
of chance at Gresham College.
In a certain 18,600 trials the distribution of frequency was
759 cases of occurrence,
3277
V
1
»
5607
>>
2
Dccurrences,
5157
>J
3
>>
2701
J)
4
»
907
»
5
it
165
JJ
6
>>
24
;■)
7
•»
1
r>
8
j>
)?
9
55
»»
10
?»
Taking moments round the point corresponding to three
occurrences I find
/V= 501,4516, /V= 8433,6021,
/*,'= 1815,5376, /*,'= 10504,6774.
^/= 1948,5484
244 Prof. K. Pearson on certain Properties
Thus the mean is :
2498,5484;
and transferring moments to this mean, we have
^2 = 1564,0839,
ya 3 = 530,4806,
^=7*074,6464,
^ = 7903,2620;
and
/3 X = 073,5460,
&= 2891,9091,
/3 3 = 9'525,2597.
Substituting in (32) we find
n = 65203,378.
Hence by (33)
^ = 451,811067,
and (34)
^ = 28391404.
Thus
S 2  28391404?+ 451,811067 = 0.
This leads to
m 2 = 1692229,
6 = 26699175.
Whence by (36)
mj=— 2685115.
Thus (37) is now
? 2 + 2685115f+1692229,
and
a= 1010546,
/3= 1674569.
Then from (40) we find
c=972077,
and from (41)
d=2'5229.
Thus we conclude that the frequency may be represented
by a hypergeometrical series of which the start is *0244 before
zero occurrence, the base unit is '9721, and the mean is at
24985. Further, from (39)
r= 101055,
jt> = 7432, £ = 2568;
or
j9tt=484577, ^=167457.
Further, we conclude that the range .of frequency cannot be
of the Hypergeometrical Series, 245
greater than 10, and the whole distribution might be closely
represented by drawing 10 balls 18,600 times out of a bag
containing 17 white and 48 black balls, and counting the
white occurrences in each draw. Actually the frequency was
obtained by drawing 10 cards out of an ordinary pack of 52
and counting the hearts in each draw. Thus we have : — ■
Actually. From theory.
Start at '0244
Mean 25 2 4988
Number drawn .. . 10 101055
Base unit 1 '9721
p 75 7432
q '25 2568
n 52 652034
Now it is clear that the first six results are in good agree
ment, but that n diverges from its actual value by 25 per cent,
although the number of trials, 18,600, is far larger than are
recorded in most practical cases.
It is of interest to record the actual and theoretical
frequencies : —
No. of hearts. Observed. Theory.
■0 ...... 759 7475
1 3277 3239
2 5607 5642
3 .. .. 5159 5172
4 ...... 2701 2743
5 907 87L
6 165 166
7 24 185
8 1 1
y o o
10 : o o
The deviations are four positive and four negative, and
four above and four below their respective probable errors.
Thus the experimental results are in good accordance with
theory.
Notwithstanding this, n has a large deviation from its
theoretical value when determined by moments. It is clearly
a quantity, when thus determined, liable to very large probable
error. Thu«, while the problem is theoretically fully solved —
ana 1 it is difficult to believe that any other solution can have
less probable error — yet we meet, unless we take an immense
number of trials, with large variations in our estimate of the
number from which the drawing is made. I have tested this
Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. S
246 Lord Rayleigh 071 James BernouilU's
on a variety of series in games of chance and on biostatis
tical data, — a small change in a high moment makes a large
change in n. Accordingly we are liable to form quite
erroneous impressions of the nature of the hypergeometrical
series, and even to reach impossible values for p, g, and r 1
which are determined through n. Thus the problem, which
is practically an important one, as enabling us to test the
sufficiency of the usual hypothesis, n = c© , of the theory
of errors, i. e. to test the " independence or interdependence
of contributory causes," is seen to admit of a solution, but
one which is hardly likely to be of much service unless in the
case to which it is applied a very large amount of data is
available.
XVI. On James Bernoulli? s Theorem in Probabilities.
By Lord Rayleigh, F.R.S*
IF p denote the probability of an event, then the probability
that in jjl trials the event will happen m times and fail n
times is equal to a certain term in the expansion of (p+qY>
namely,
■A^.P^", (i)
where p+q = l, m + n=r/ji.
" Now it is known from Algebra that if m and n vary
subject to the condition that mf n is constant, the greatest
value of the above term is when m/n is as nearly as possible
equal to p/q, so that m and n are as nearly as possible equal
to fip and fxq respectively. We say as nearly as possible,
because /xp is not necessarily an integer, while m is. We
may denote Ihe value of m by pp + z, where z is some proper
fraction, positive or negative ; and then n = pq—z"
The rth term, counting onwards, in the expansion of
{p+qY after (1) is
**!
m
? , t p m ~ r q n+r  > . . (2)
r I n j r ! v '
The approximate value of (2) when m and n are large
numbers may be obtained with the aid of Stirling's theorem,
viz.
^! =/ ^*^27r)£L+ JL + ...J., . (3 )
* Communicated bv the Author,
Theorem in Probabilities. 247
The process is given in detail after Laplace in Todhunter's
•History of the Theory of Probability/ p. 549, from which
the above paragraph is quoted. The expression for the rth
term after the greatest is
.,r2
v^ j~ fxrz r(n — ni) r z ? 3 \ ,.
mf\ + »w i ~2mJi bW + 6V J *' ' *' '
</{27rm
and that for the rth term before the greatest may be deduced
by changing the sign of r in (4).
It is assumed that r 2 does not surpass fi in order of mag
nitude, and fractions of the order l//x are neglected.
There is an important case in which the circumstances are
simpler than in general. It arises when p = q^ J, and /jl is
an even number, so that m = /z — J/*. Here z disappears
ab initio, and (4) reduces to
P
representing (2), which now becomes
(6)
An important application of (5) is to the theory of random
vibrations. If /x vibrations are combined, each of the same
phase but of amplitudes which are at random either f 1 or — 1,
(5) represents the probability of ifi> + r of them being positive
vibrations, and accordingly \y^— r being negative. In this
case, and in this case only, is the resultant + 2r. Hence if x
represent the resultant, the chance of #, which is necessarily
an even integer, is
2e* 2 /V
n/(27T/*)'
The next greater resultant is (^' + 2); so that when a is
great the above expression may be supposed to correspond to
a range for x equal to 2. If we represent the range by dx,
the chance of a resultant lying between x and x + dx is given by
e x2 ^dx
s/(?tth) {)
Another view of this matter, leading to (5) or (7) without
the aid of Stirling's theorem, or even of formula (1), is given
* Phil. Mag, vol. x. p. 75 (1S80).
218 Lord Rayleigh on James Bernoulli? 's
(somewhat imperfectly) in ' Theory of Sound,' 2nd ed. § 42 a.
It depends upon a transition from an equation in finite differ
ences to the wellknown equation for the conduction of heat
and the use of one of Fourier's solutions of the hitter. Let
J(fjL,r) denote the chance that the number of events occurring
(in the special application positive vibrations) is \p. ■% r, so
that the excess is r. Suppose that each random combination
of /jl receives two more random contributions — two in order that
the whole number may remain even, — and inquire into the
chance of a subsequent excess r, denoted by/(/x + 2, r). The
excess after the addition can only be r if previously it were
r — 1, r, or r\l. In the first case the excess becomes r by
the occurrence of both of the two new events, of which the
chance is J. In the second case the excess remains r in
consequence of one event happening and the other failing,
of which the chance is J ; and in the third case the excess
becomes r in consequence of the failure of both the new
events, of which the chance is 5. Thus
fQ* + i,r)=if{p,rl) + Vfar) + lfOhr + I). . (8)
According to the present method the limiting form of /is to be
derived from (8). We know, however, that/ has actually the
value given in (6), by means of which (8) may be verified.
Writing (8) in the form
we see that when /x and r are infinite the lefthand member
becomes 2df/d^i, and the righthand member becomes ^d 2 f/dr 2 ,
so that (9) passes into the differential equation
dfi 8 dv l ••••••♦ \m
In (9), (10) r is the excess of the actual occurrences over \yu.
If we take x to represent the difference between the number
of occurrences and the number of failures, x=2r and (10)
becomes
iLl c ll . . . (in
In the application to vibrations f(fi, x) then denotes the
chance of a resultant + o? from a combination of ja unit
vibrations which are positive or negative at random.
In the formation of (10) we have supposed for simplicity
that the addition to /x. is '2, the lowest possible consistently
with the total number remaining even. But if we please we
may suppose the addition to be any even number yJ , The
Theorem in Probabilities. 249
analogue of (8) is then
+ ^f^ ) /(^''^' + 2)+..+/(^'' + ^');
and when ^ is treated as very great the righthand member
becomes
/(„,,.) {i+^'+^^ '+...+/+ 1}
+ +/( /i '2) 2 + l./ i ' 2 }
The series which multiplies / is (L + l)% or 2^. The
second series is equal to /u' . 2^', as may be seen by com
parison of coefficients of x l in the equivalent forms
(e x + e x ) n =2 n (l + %a: 2 + . . .) n
= 6»* + 7^" 2 > + ^"""^ *(—*)* + ....
I . w
The value of the lefthand member becomes simultaneously ■
so that we arrive at the same differential equation (10) as
before.
This is the wellknown equation for the conduction of heat,
and the solution developed by Fourier is at once applicable.
The symbol fi corresponds to time and r to a linear co
ordinate. The special condition is that initially — that is when
fi is relatively small — f must vanish for all values of r that are
not small. We take therefore
/(ft*) = ^***, (12)
which may be verified by differentiation.
The constant A may be determined by the understanding
that/(yu, r) dr is to represent the chance of an excess lying
between r and r + dr, and that accordingly
f + 7(ft r)dr=l. ..... (13)
*/ — °0
e ~ z ~dz= s/tt, we have
&~V(h> ■•■■■ (U >
250 On James Bernoulli? s Theorem in Probabilities.
and, finally, as the chance that the excess lies between r and
r + dr,
\Z&^ ( is )
Another method by which A in (12) might be determined
would be by comparison with ((3) in the case of ?' = (). In
this way we find
A
pi
1.3.5.
I~ 2.4.6,
by Wallis' t
...p1
s/fi
2*. tr I &
= \/fc)
heorem.
If, as is natural in the problem of random vibrations, we
replace r by x, denoting the difference between the number of
occurrences and the number of failures, we have as the chance
that x lies between x and x + dx
(16)
n/(2tt/*)'
identical with (7).
In the general case when p and q are not limited to the
values J, it is more difficult to exhibit the argument in a
satisfactory form, because the most probable numbers of
occurrences and failures are no longer definite, or at any rate
simple, fractions of ft. But the general idea is substantially
the same. The excess of occurrences over the most probable
number is still denoted by r, and its probability by f(fi, r).
We regard r as continuous, and we then suppose that /j,
increases by unity. If the event occurs, of which the chance
is p, the total number of occurrences is increased by unity.
But since the most probable number of occurrences is increased
by p, r undergoes only an increase measured by 1 — p or q.
In like manner if the event fails, r undergoes a decrease
measured by p. Accordingly
f(n+%r)=pf{n r rq) +qffar+.p). . . (17)
On the right of (17) we expand f{p,r—q),f(/j,, r+p) in
powers of p and q. Thus
Notices respecting New Books. 251
so that the righthand member is
O + q)f+ i j^ (p 2 q +pq*), or /+ \ n ^.
The lefthand member may be represented by f+df/dp,, so
that ultimately
df 1 ffif .,.#.
d^=^d? (1&)
Accordingly by the same argument as before the chance of
an excess r lying between r and r {dr is given by
— i ^e^P^dr (19)
We have already considered the case of p = q= z i' Another
particular case of importance arises when p is very small,
and accordingly q is nearly equal to unity. The whole
number /j, is supposed to be so large that pp,, or m, repre
senting the most probable number of occurrences, is also
large. The general formula now reduces to
,<l \ e* 2 ' 2m dr, (20)
V (257T/H)
which gives the probability that the number of occurrences
shall lie between m + r and m + r+dr. It is a function of m
and r only.
The probability of the deviation from m lying between +r
"■ '/ fl i [ r e^dr=  f T '—**> ' ' ' ( 21 )
y/ (Jtirm). Jo vttJo
where T = r/ N /(2m). This is equal to "84: when t=1*0, or
r — s/{2m) ; so that the chance is comparatively small of a
deviation from m exceeding + s /(2m). For example, if m
is 50, there is a rather strong probability that the actual
number of occurrences will lie between 40 and 60.
The formula (20) has a direct application to many kinds of
statistics.
XVII. Notices respecting New Books.
Textbook of Algebra with exercises for Secondary Schools and Colleges.
By G. E. Fisher, M.A., Ph.D., and I. J. Schwatt, Ph.D. Part I.
(pp, xivf683: Philadelphia, Fisher & Schwatt, 1898).
r PHlS is a big book for the comparatively small extent of ground
** it covers. The usual elementary parts are discussed up to
and including simultaneous Quadratic equations, and then, in the
remaining 80 pages, we have an account of liatio, Proportion,
Variation, Exponents, aud Progressions. The Binomial Theorem
for a positive Integral Exponent occupies about a dozen pages, the
treatment by Combinations being reserved, we presume, for Part 11.
252 Intelligence and Miscellaneous Articles.
The text has been very carefully drawn up and should be useful
for young teachers. The distinction between the signs of operation
and the signs of quality is very clearly indicated by means of a
special notation. There is a good chapter on the interpretation
of the solutions of Problems, such questions as that of the problem
of the couriers and allied problems being worked out in some
detail. In some few places one would have expected the Authors
to have been a little fuller, but the general level is high. The
exercises are very numerous and well graded. The number is
intentionally large " that the teacher from year to year may have
Variety with different classes." There is no mention of Graphs, a
branch of the subject which Prof. Chrystal's book brings into
prominence. The book can be resommend^d as a sound treatise
on the elements of Algebra, and the printers have done their work
well. The most important typographical error we have come,
across is on p. 611 line 5, where for " first" read " second." There
are no answers at the end.
XVIII. Intelligence and Miscellaneous Articles.
RELATIVE MOTION OF THE EARTH AND .ETHER.
To the Editors of the Philosophical Magazine.
Gentlemen,
IN your September number Dr. Lodge comments on my objection
to the conclusiveness of the MichelsonMorley aether experi
ment, and I should like to point out that his remarks are founded
on a strange misconception of the nature of my objection. He
conjectures that I attribute the negative result of the Michelson
and Morley experiment " to the possible secondorder influence of
a hitherto neglected firstorder tilting or shifting of the wavefronts
brought about by the undiscovered drift of the aether past the
earth." But in my communication I pointed out that nay objection
was not of this nature, but related to the assumption made as to
the optical sensitiveness of the system of interferencefringes
relied ou by the experimenters to enable them to measure the
minute length in question in their experiments. My contention
was simply that the system of fringes used in the experiment had
probably a more complex character than was supposed, and that
therefore its capability of measuring the small length accurately
was overrated to an unknown extent. Evidently Dr. Lodge has
pondered so deeply on aberration problems that in reading my
paper his thought has got into sone old groove which he has
unconsciously taken to be the direction of my argument.
Yours obediently,
William Sutherland.
Postscript by Prof. Lodge. — I was not very clear about Mr;
Sutherland's precise line of argument, nor am I now ; but there
was an imaginary loophole which others might attempt to get
through, though Mr. Sutherland, as it now appears, did not; and
I took the opportunity (not specially opportune as it turns out) of
indicating that it was closed.
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[FIFTH SERIES.] J^V"
.U. WM
MARCH 1899. V. „ r fpV
XIX. On the Theory of the Conduction of Electricity through
Gases by Charged Ions. By J. J. Thomson, M.A., F.R.S.,
Cavendish Professor of Experimental Physics, Cambridge*.
rilHE electrical conductivity possessed by gases under cer
JL tain circumstances — as for example when Rontgen or
uranium rays pass through the gas, or when the gas is in a
vacuumtube or in the neighbourhood of a piece of metal
heated to redness, or near a flame or an arc or sparkdischarge,
or to a piece of metal illuminated by ultraviolet light — can be
regarded as due to the presence in the gas of charged ions,
the motions of these ions in the electric field constituting the
current.
To investigate the distribution of the electric force through
the gas we have to take into account (1) the production of
the ions ; this may either take place throughout the gas, or
else be confined to particular regions ; (2) the recombina
tion of the ions, the positively charged ions combining with
the negatively charged ones to form an electrically neutral
system ; (3) the movement of the ions under the electric
forces. We shall suppose in the subsequent investigations
that the velocity of an ion is proportional to the electric
intensity acting upon it. The velocity acquired by an ion
under a given potential gradient has been measured at the
Cavendish Laboratory by several observers — in the case of
gases exposed to the Rontgen rays by Rutherford and
by Zeleny; for gases exposed to uranium radiation or to
* Communicated by the Author.
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. T
254 Prof. J. J. Thomson on the Theory of Conduction of
ultraviolet light by Rutherford ; for the ions in flames by
McClelland and H. A. Wilson; and for the ions in gases
near to incandescent metals or to the arc discharge by
McClelland. The velocities in the different cases vary very
much ; the velocity of an ion in the same gas is much the
same whether the conductivity is due to Rontgen rays,
uranium rays, or ultraviolet light; it is much smaller when
the conductivity is produced by an arc or by incandescent
metal. Thus the mean velocity of the positive and negative
ions under a volt per centimetre in air exposed to Rontgen
rays was found by Rutherford to be about 1'6 cm./sec, while
for gas drawn from the neighbourhood of an arcdischarge
in carbonic acid the mean velocity of the positive and negative
ions was found by McClelland to be only "0035 cm./sec.
This difference is caused by the ions acting as nuclei about
which condensation, whether of the gas around them or of
watervapour present in the gas, takes place. The power
of these ions to act as nuclei for the condensation of water
vapour is strikingly shown by C. T. R. Wilson's * experi
ments on the effects of Rontgen and uranium radiation on
the formation of clouds, and also by R. v. HelmhohVsf
experiments on the effects produced by ions on a steam jet.
If the size of the aopreoation which forms round the ion
depends on the circumstances under which the ion is liberated
and the substances by which it is surrounded, the velocity
which the ion acquires under a given potential gradient will
also depend on these circumstances, the larger the mass of
the aggregation the smaller will be this velocity. A remark
able result of the determination of the velocities acquired by
the ions under the electric field is that the velocity acquired by
the negative ion under a given potential gradient is greater
than (except in a few exceptional cases when it is equal to)
the velocity acquired by the positive ion. Greatly as the
velocities of the ions produced in different ways differ from
each other, yet they all show this peculiarity. The relative
velocities of the negative and positive ions differ very much
in the different cases of conduction through gases ; thus in
the case of imperfectly dried hydrogen traversed by the
Rontgen rays, Zeleny found that the speed of the negative
ions was about 25 per cent, greater than that of the positive,
while in the case of conduction through hot flames H. A.
Wilson found that the velocity of the negative ion was 17 or
18 times that of the positive. In the case of the discharge
through vacuumtubes, the measurements which I made of
* Wilson, Phil. Trans. A, 1897 ; Proceedings of Cambridge Phil. Soc.
vol. ix. p. 333.
t R. v. Helmholtz, Wied. Ann. vol. xxvii. p. 509 (1886).
Electricity through Gases by Charged Ions. 255
tbe ratio of the charge to the mass for the particles con
stituting the cathode rays and those of W. Wien * for the
ions carrying the positive charge indicate that the ratio of
the velocity of the negative ion to that of the positive one
under the same potential gradient would be very large.
This fact is, I think, sufficient to account for most of the
differences between the appearances at the positive and nega
tive electrodes in a vacuumtube. Schuster (Proc. Roy. Soc.
vol. xlvii. p. 526, 1890), from observations on the rates at which
positively and negatively electrified bodies lost their charges
in a vacuumtube, came to the conclusion that the negative
ions diffused more rapidly than the positive ; other pheno
mena connected with the discharge led me later (Phil. Mag.
vol. xl. p. 511, 1895) independently to the same result.
We shall now proceed to find equations satisfied by the
electric intensity in a gas containing charged ions. To simplify
the analysis we shall suppose that the electric force is every
where parallel to the axis of x, and that if X is the value of
the electric intensity at a point, the velocity of the positive ion
at that point is &iX, and that of the negative ion in the
opposite direction & 2 X; we shall suppose that at this point the
number of positive ions per unit volume is %, the number of
negative ions n 2 ; let q be the number of positive or negative
ions produced at this point in unit volume in unit time : the
number of collisions per unit time between the positive and
negative ions is proportional to n x ii 2 . We shall suppose that
in a certain fraction of these collisions recombination between
the positive and negative ions takes place, so that a number
un\n 2 of positive and negative ions disappear in unit time
from unit volume in consequence of the recombination of the
ions. If e is the charge carried by each ion, the volume
density of the electrification is (n x — n 2 ) e, hence we have
_ =A7r(n 1 ^n 2 )e ! , (1)
if i is the current through unit area of the gas, and if we
neglect any diffusion except that caused by the electric field,
^n^X + Ji 2 n 2 eK. = i, (2)
and if things have settled into a steady state, i is constant
throughout the gas; from these equations we have
1 f l , k 2 afX  ....
" ie= /tT+I 2 lx + i^j' • • • ( 3 )
* W. Wien, Verhandl. der phys. GeseUsch. zu Berlin, vol. xvi. p. 165.
T2
256 Prof. J. J. Thomson on the Theory of Conduction of
In a steady state the number of positive ions in unit volume
at a given place remains constant, hence
j
^(k 1 n 1 X)=q—an 1 n 2 , (5)
and
— — (k 2 n 2 X) =9. — *n x n 2 .
Substituting in either of these equations the values of n Y n 2
previously found we get, since di/d,v = 0,
lire k^k.dA lx)~ 9 XVft + ifc^V 4tt rfar/V ** dx t
If we put X 2 = 2y,
this equation becomes
1 ^1^2 ^P * i h \/ ___h \
Aire^ + k, P dy ~ q 2ye' 2 (k l + k 2 ) 2 \ ±ir P )\ ^ P )'
I have not been able to integrate this equation in the
feneral case when q is finite and k x not equal to k 2 . We can,
owever, integrate it when q is constant and k x =:k 2 = k. In
this case the equation may be written
77
d I k * 2 2 \ « 1/ ^ 2 gV J.
the solution of which is
where C is a constant of integration.
If the current through the gas passes between two parallel
plates maintained at a constant potentialdifference, d~K/dx =
midway between the plates ; at the positive plate n x — 0, while
w 2 = at the negative plate; hence if X , X x be respectively
the values of X midway between the plates and at either plate,
we have, putting p = 0, to get X
qek
$7rek
Electricity through Gases by Charged Ions. 257
But
fc=2n&eX ,
since when dX/dx = 0, n 1 =n 2 .
The measurements of X for gases exposed to Rontgen rays
show that unless the current is approaching the maximum value
it can attain, X is practically constant for some distance near
the middle of the plates; hence in this case we have d 2 X/d% 2 =
midway between the plates, and therefore by equations (3)
and (5) q=a?i 2 ; substituting this value of n we have
Swek'
or
X* — =CXo= (6)
1
Sirek
k~
At either plate ^2 = 0, so that JQ2P 2 — * 2 = 0, thus
gek
X?_J^ =CX 1 i^ ; .... (7)
1
a
Sirek
hence from (6) and (7)
.„ Sirek
or it
 i =/5 2 "V (8)
It follows from this equation that X ly /X is greater than
unity, and that the value of this ratio increases from unity to
infinity as /3 increases from zero to infinity. We see that /3
does not involve either q or i. So that, to take a particular
case, when the gas between two plates is exposed to Rontgen
rays, the ratio of the electric intensity at the plates to that
midway between them is independent of the intensity of the
radiation and of the current through the gas. The curves
258 Prof. J. J. Thomson on the Theory of Conduction of
giving the connexion between the electric intensity and the
distance between the plates are found by experiment to be
somewhat as represented in fig. 1. The variation in X (fig. 1)
Fisr. 1.
occurs only in two layers near the plates and X is approximately
constant in the rest of the field. As the current through the
gas increases the layers of inconstant X expand until they
touch, and then there is no longer a region in which X is
constant. We can easily find an inferior limit to the value
of X the thickness of one of these layers when we are given
the value of the current. For suppose P (fig. 1) is at the
boundary of the layer next the positive electrode, then at P,
since X becomes constant, half the current must be carried
by the positive and half by the negative ions; if i is the
current and e the charge carried by an ion, then ij'le positive
ions must cross unit area of a plane through P in unit time,
so that at least that number must be produced in unit time in
the region between P and the positive plate. Now if X is
the thickness of the layer, qX is the number of positive ions
produced in unit time; the number that cross the plane in
unit time cannot then be greater than qX, and will only be as
great as this if no recombination of the ions takes place ;
hence
or
X>
2e :
2qe
Thus t/2eq is an inferior limit to X. It will not, however, I
think be very far from the true value, for we can show that
but little recombination will take place in the time taken by
the positive ions to traverse a layer of this thickness. For
the rate of combination of the positive ions is given by
dn x
= — an l ?i 2 .
Electricity through Gases by Charged Io?is. 259
If N 2 is the maximum value of n 2 , then ^= measures the time
that is taken before recombination diminishes the number of
the ions to any very appreciable extent. In this time the
positive ions would move a distance 8 given by the equation
where X t is the value of X at the plate.
&x 2 x,<?=^
thus
1
Let X x =7X0 where 7 = /3 2  2/ < J ,
^ yWX*e
(XI
r^_7 2 x
±qe 2
Thus if 7 is tolerably large, the positive ions will traverse
a space much greater than A, before recombining.
The greatest current that can pass between the plates is
when all the ions are used in carrying the current; if I is the
distance between the plates, then Iq positive and negative
ions are produced in unit time ; thus if I is the maximum
current which can pass between the plates
hence we can write
I = lqe;
\ t
7~2l'
The equations
dX
,^=47r(tt 1 n 2 )^
^(k 1 7i l X)=qan 1 n 2 ,
J x {k 2 n 2 X) = — (q — ocn^),
(k 1 n 1 + k 2 n 2 )Xe = c }
260 Prof. J. J. Thomson on the Theory of Conduction of
are satisfied by
7i l = n 2 = {q/a}i y
k 2 n 2 ~K.i
h
©'
^l + A2 '
e{k x + kc.
where c is the current through the gas. In this case the
amounts of the current carried by the positive and negative
ions respectively are proportional to the velocities of those
ions. When, however, the current passes between two
parallel plates, this solution cannot hold right up to the plates,
For, consider the condition of things at a point P, between
the plates AB and CD, of which AB is the positive and CD
the negative plate. Then across unit area at P
k, i
k,+ko e
positive ions pass in unit time, and these must come from the
region between P and AB ; this region can, however, not
furnish more than q\, and only as much as this if there are
no recombinations ; hence the preceding solution cannot hold
when the distance from the positive plate is less than
k x + k 2 qe '
similarly it cannot hold nearer the negative plate than the
distance
k 2 i
k x + k 2 qe
We shall assume that the solution given above does hold in the
parts of the field which are further away from the plates than
these distances ; and further, that in the layers in which this
solution does not hold, there is no recombination of the ions.
Let us now consider the condition of things near the positive
plate between
x = and x — , — ^ = X say.
k x + k 2 qe
Then, since in this region there is no recombination, our
Electricity through Gases by Charged Ions. 261
equations are
— = 4:7r{n 1 n 2 )e,
^{k 2 n 2 X)=q.
If q is constant, we have
A: 1 ?z 1 X = got,
# 2 ?2 2 X= go:,
the constant has been determined so as to make ni~0 when
x=0 ; substituting these values for n 1? n 2 , in the equation
giving dUL/dx we get
or
,2
x Kf(H,)f>^
the constant may be determined from the condition that
when x = \
^~qe\k 1 + k 2 ) 2J
from this we find
~ OC L 2 f , 47T6 £. ._ _ . 1
Since C is the value of X 2 when w=0, it is the value of X 2
at the positive plate; if we call Xj the value of X at the positive
plate and X the value of X between the layers we have
w{it^(M*>}'.
thus X x is always greater than X , the value of X between the
layers.
If X 2 is the value of X at the negative plate we have
Thus, if k 2 , the velocity of the negative ion, is very large
262 Prof. J. J. Thomson on the Theory of Conduction of
compared with k 1} the velocity of the positive, the value of
X at the negative plate is large compared with its value at the
positive.
The curve representing the electric intensity between the
plates is shown in jSg. 2. In this case & 2 /^i is a large
quantity.
Fig. 2.
The fall of potential across the layer whose thickness is X x
is equal to
and this is equal to
1 !Ldx.
Jo
IXqXj
where
and
^X 1 X 1 +i^log(v/ / 3+ V1 + /3):
Hence, if fi is large, the fall of potential across the layer
whose thickness is \j next the positive plate, is approximately
Similarly, if
and if /Si is large the fall of potential across the layer whose
thickness is X 2 is approximately
Electricity through Gases by Charged Ions. 263
The distance between the plates in which the electric
intensity is constant and equal to X is I— {\ t +\), where / is
the distance between the plates. Since \ l + \ 2 = i/qe tne ^ a ^
of potential in this distance is equal to
hence if V is the potentialdifference between the plates
and
X = ^V l
^° \q) efa + ki)'
so that
\q/ cft + y I v k Y + k 2 qe JV k x +k 2 qe qe J
This gives the relation between the current and the poten
tialdifference between the plates. It is of the form
y = A* 2 +B t .
In a paper by Mr. Eutherford and myself in the Phil.
Mag. for Oct. 1896, a relation between V and i was given on
the assumption that the electric intensity was constant between
the plates ; in this investigation I have tried to allow for the
variation in the electric intensity. The above investigation
ceases to be an approximation to the truth when the two
layers touch each other. In this case the current has its
limiting value Iqe, and there is no loss of ions by recom
bination ; we may therefore neglect the recombination and
proceed as follows.
Equations (5) become in this case
^ (k 1 n 1 X) = q,
^(k 2 n 2 X)=q.
If q is constant, the solutions of these equations are
Ar 1 w 1 X = ga?, (9)
k 2 n 2 X=zq{lx), (10)
where x is the distance measured from the positive plate and
I the distance between the plates, for these solutions satisfy
264: Prof. J. J. Thomson on the Theory of Conduction of
the differential equations and the boundary conditions »!=<)
when # = 0, and ?i 2 = when x—l. From the equation
— =±7r(? h n 2 )e,
we have
or
where C is the constant of integration. When X has its
minimum value we see from equation (1) that n^^ — ^; hence
from (9) and (10) at such a point we have
k x x
irtt ( 12 >
hence if we determine the point Q where X is a minimum, this
equation will give us the ratio of the velocities of the positive
and negative ions.
We see that a positive ion starting from the positive
plate, and a negative ion starting from the negative plate,
reach this point simultaneously.
If X is the minimum value of X, and f the distance of a
point between the plates from Q, we may write equation (11)
in the form
if I is the maximum current, this may be written
xx^+w^I+i). . . . {13)
We see from this equation that if we measure the values of
X at two points and I, the maximum current, we can deduce
the value of
and since from (12) we know the value of k v >k 2 , we can
deduce the values of k ± and k 2 .
If the positive ion moves more slowly under a given
potential gradient than the negative ion, then we see *from
(12) that Q is nearer to the positive than to the negative
Electricity through Gases by Charged Ions. 265
plate. Hence it follows from (13) that tho electric force at
the negative plate is greater than that at the positive.
A very convenient method of determining the velocities of
the ions, and one which can be employed in nearly every case
of conduction through gases, is to produce the ions in one
region and measure the electric intensity at two points in a
region where there is no production of ions, but to which ions
of one sign only can penetrate under the action of the
electric field. Thus let A, B represent two parallel plates
immersed in a gas, and let us suppose that in the layer
between A and the plane LM we produce a supply of ions,
whether by Rontgen rays, incandescent metals, ultraviolet
light, or by other means, and suppose that the gas between
LM and B is screened off from the action of the ionizer.
Then if A and B are connected to the poles of a battery a
current will pass through the gas, and this current in the
region between LM and B will be carried by ions of one
sign. These will be positive if A is the positive pole, nega
tive if it is the negative pole. Let us find the distribution
of electric intensity in the region between LM and B. Let
us suppose A is the positive plate, then all the ions in this
region are positive and we have, using the same notation as
before,
dX A
k^Xe = t,
where i is the current through unit area ; from these equations
we have
^ dX _ 4ttc
dx ~~ k l '
or
h
Hence if we measure the values of X at two points in
the region between LM and B, and also the value of t,
we can, from this equation, deduce the value of k u and
hence the velocity of the positive ion in a known electric
field. To determine the velocity of the negative ion we
have only to perform a similar experiment with the plate A
negative.
When the ionization is confined to a layer CD between the
plates A and B, the distribution of electric intensity is repre
266 Prof. J. J. Thomson on the Theory of Conduction of
sented by fig. 3, where A is the positive and B the negative
plate, and the velocity of the negative ion is supposed to be
much greater than that of the positive.
Ksr. 3.
The investigation of the distribution of electric intensity
given on p. 262 shows that when the velocity of the negative
ion is much greater than that of the positive, the distribution
of the intensity has many features in common with that
associated with the passage of electricity through a vacuum
tube, especially the great increase in electric intensity close
to the negative electrode. Thus this feature of the discharge
through vacuumtubes can be explained by the greater
velocity of the negative ion than of the positive, a property
which seems to hold in all cases of discharge of electricity
through gases. And as the most important of the differences
between the phenomena at the two poles of a vacuumtube
are direct consequences of the electric intensity at the cathode
far exceeding that at the anode, I think the most striking
features of the discharge through vacuumtubes are conse
quences of the difference in velocity between the positive and
negative ions. In the case discussed on p. 261 we assumed
q constant, i. e. that the ionization along the path of the
discharge was constant ; in the case of the discharge through
vacuumtubes, where the ionization is due primarily to the
electric field itself, it is unlikely that the ionization will be
constant when the field is so variable. We can derive
information as to the distribution of the ionization by a study
of the very valuable curves giving the distribution of electric
intensity in a vacuumtube which we owe to the researches
of Graham (Wied. Ann. lxiv. p. 49, 1898).
From the equations
_= 4:7r(n l n 2 )e,
Electricity through Gases by Charged Ions. 207
cl
dx
(k^i^X) =q — an 1 n 2)
 X. (^ 2 X) =^«??in 2 ,
we get if k 1 and &o are independent of x
cf~X 2
(/*'"'
S7re{qa ni n 2 )(^ + jJ.
Thus q — a.n Y n. 2 is of the same sign as d 2 X. 2 /dx 2 . Tims when
^ — a/^/^2 is positive, that is when the ionization exceeds the
recombination, the curve for X 2 will be convex to the axis
of x, and this curve will be concave to the axis of x when the
recombination exceeds the ionization. Places of sharp cur
vature will be regions either of great ionization or recom
bination. Fig. 4 is a curve for X 2 calculated from Graham's
Fig. 4.
Hi
results. It will be seen that there arc two places of specially
sharp curvature with the curvature in the direction denoting
ionization, one, the most powerful one, just outside the
negative dark space, the other near the anode, while in the
positive light the curvature indicates recombination. It
would seem as if the positive ions formed at the centre of
268 Conduction of Electricity through Gases by Charged Ions.
ionization near the anode, in travelling towards the cathode,
met with the negative ions coming from the centre of
ionization near the cathode, that these positive ions combine
with the negative nntil their number is exhausted, and on
combining give out light, the region of recombination con
stituting the positive light. In the dark space between the
positive light and the negative glow these positive ions from
the centre of ionization near the anode are exhausted, so that
there are none of them left for the negative ions coming from
the centre near the cathode to combine with.
The nick in the curve denoting the centre of ionization
near the cathode is present in all the curves given by
Graham ; the centre near the anode is not nearly so per
sistent. In several of the curves given by Graham there is
no nick near the anode, though the one near the cathode is
well marked, and in these tubes there is no welldeveloped
positive light. The distribution of potential which accom
panies the luminous discharge requires a definite distribution
of electrification in the tube, this requires ionization and a
movement of the ions in the tube before the luminous dis
charge takes place. There must, therefore, be a kind of
quasidischarge to prepare the way for the luminous one.
Warburg ( Wied. Ann. lxii. p. 385) has, in some cases, detected
a dark discharge before the luminous one passes. It seems
probable that such a discharge is not limited to the cases in
which it has already been detected, but is an invariable
preliminary to the luminous discharge.
Besides the " nicks " or places of specially sharp curvature
fig. 4 shows that there is a small curvature in the direction
indicating an excess of ionization over recombination all
through the considerable space that intervenes between the
positive light and the negative glow ; as this region is one far
away from places of great electric intensity it seems probable
that in producing ionization the electric intensity at any
point is helped by other agencies. The case of the cathode
rays shows that the motion of charged ions tends to ionize
the surrounding gas. E. Wiedemann, too, has shown that
the discharge generates a peculiar radiation, called by him
a Entladungstrahlen"; it is possible that these may possess
the power of ionizing a gas through which they pass.
[ 269 ]
XX. Cathode, Lenard, and Rontgen Rays.
By William Sutherland *.
^PO explain the results of his experiments on cathode rays,
* and to account for the HertzLenard apparent passage
of cathode rays through solid bodies according to Lenard's
wonderfully simple law, J. J. Thomson (Phil. Mag. [5] .
xliv., Oct. 1897) proposes the hypothesis, that the matter in
the cathode stream consists of atoms resolved into particles
of that primitive substance out of which atoms have been
supposed to be composed. Before a theory of such
momentous importance should be entertained, it is necessary
to examine whether the facts to be explained by it are not
better accounted for by the logical development of established
or widely accepted principles of electrical science.
The chief facts which Thomson arrives at from his experi
ments are : — That the cathode rays travel at the same speed
in different gases such as hydrogen, air, and carbonic dioxide;
and that m/e, the ratio of the mass of the particles to their
charge, is the same for the cathode streams in all gases, and
is about 10 3 of the ratio of the mass of the hydrogen atom
to its charge in ordinary electrolysis. These seeming facts
have also been brought out with great distinctness in the
experiments of Kaufmann (Wied. Ann. lxi. and lxii.).
Whatever proves to be the right theory of the nature of the
cathode rays, the quantitative results which these experi
menters have obtained (as did also Lenard), in a region,
where, amid a bewildering wealth of qualitative work, the
quantitative appeared as if unattainable, must constitute a
firm stretch of the roadway to the truth.
Let us briefly consider the theories used by J. J. Thomson
and by Kaufmann to interpret their experiments. For instance,
Thomson considers N particles projected from the cathode,
each of mass ra, to strike a thermopile, to which they give up
their kinetic energy Nmv 2 measured as W. Each of the
particles carries its charge of electricity e, the whole quantity
Ne being measured as Q. Thus we have
±v 2 m/e = W/Q (1)
But again, the particles, after being projected through a slit
in the anode with velocity v, are subjected to a field H of
magnetic force at right angles to the direction of motion, so
that the actual force tending to deflect each particle is Hev
at right angles to H and v. The result is that each particle
describes a circular path of radius p with the centrifugal
* Communicated by the Author.
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. U
270 Mr. W. Sutherland on
force mv 2 /p equal to H^v, and therefore our second equation
is
vm/e=*Rp (2)
By measuring W/Q and Hp, Thomson is able to deduce
values of v and m/e from (1) and (2), and these are the values
which lead to his remarkable conclusions already given.
Thomson and Kaufmann control the results of this method
by a second method of experimenting, in which deflexion of
cathode rays was produced by electrostatic force, as well as
by magnetic, the forces in Thomson's experiments being
adjusted so that the deflexions in both cases were the same,
and therefore, if F is the electric force,
Fe = Rev (3)
Thus an independent measure of v is taken, and as it confirms
those made by the other method, the experimental evidence
for the simplicity of the laws of cathode motion is greatly
strengthened. But in the theory of these experiments there
is one suppressed premiss, namely, that a charge e must be
associated with some mass m. Now in following up the
ionic hypothesis as far as it w T ill go, it is our duty to use this
premiss as one of the links in the chain of reasoning ; but
when it leads us to a conclusion subversive of the ionic
hypothesis, namely, that atoms are split up into particles
having different charges from the atomic charge in electro
lysis, then we are no longer bound by the ionic hypothesis.
It may therefore be that free electrons can appear in the
sether, and that in the cathode stream the greater part of the
electricity travels as free electrons.
A systematic statement of the reasons for contemplating
the possibility of the motion of free electrons through the
gether will be given below ; but in the present connexion it
is of most importance to consider whether such electrons could
give up to the thermopile the kinetic energy measured by
Thomson. From the writings of Thomson, Heaviside, Searle,
and Morton (Phil. Mag. [5] xi., xxvii., xxviii., xli., xliv.)
we can form an idea as to what takes place when an electron
is set in motion. These writings relate to electric charges
.on conducting spheres and ellipsoids, the charges being
caused to move by the motion of the conductors; but in the
case of the free electron we cannot say that its charge is on
anything, unless a modified portion of the sether. Our
simplest plan is to regard the electron as a spherical shell of
electricity of total amount e, the radius being a.
The main effect of setting such an electron in motion by
means of some source of energy, is that electric and magnetic
Cathode, Lenard, and Rout gen Rays. 271
energy are spread into the aether with the velocity of light
V, so that when the electron has velocity u the total amount
of such electric and magnetic energy is (Searle, Phil. Mag.
xliv.)
e 2 /V. Y + u n \
If u is small compared with V this is
and taking account only of the part of this energy due to
motion, we have Heaviside's result : — 
*V/3KaV 2 or fie 2 u* /3a.
Now if the process, by which some of our store of energy
was converted into electric and magnetic forms on setting
the electron in motion, is a reversible one, then on stopping
the electron in a suitable manner the electric and magnetic
energy ought to flow back to our source or to the stopping
body, and if there are no arrangements at the stopping body
suitable for storing this as ordinary kinetic or potential
energy, it will appear as heat amongst the particles which
take part in the stoppage. Thus, then, certain actions of a
moving electron take place as if it had a localized inertia,
just as in the theory of electric currents a large part of their
behaviour is such as it would be if the moving electricity
had localized inertia. According to Searle's expression, the
inertia or effective mass of the electron becomes a function of
its velocity, if we define it as the quantity which is to be multi
plied by half the square of the velocity to give the kinetic
energy. With Heaviside's expression for smaller velocities,
we should have the inertia equal to 2[ie 2 /3a. But apart from
these details, we have only to assume that the energy
imparted to an electron when it is set in motion (or the
greater part of it) is given up as heat to the material particles
which arrest its motion, and is equal to half the square of
the velocity multiplied by a certain quantity characteristic
of the electron and appearing by the symbol m in the equations
of Thomson and Kaufinann. Then the experimental results are
at once explained ; for as the negative electrons are the same
in all the experiments, m/e has the same value for cathode
streams in all gases : the gas facilitates the electric discharge,
but does not control it ; as a steamengine can give the
same results with several lubricants, so the cathode stream
can give the same stream of electric energy by means of its
U 2
272 Mr. W. Sutherland on
moving free electrons, whatever may be the gas used to
facilitate its flowing.
We can use Thomson's and Kaufmann's value of mje,
namely, about 10~ 7 when e is measured in electromagnetic
units, to calculate the order of magnitude of a the radius of
the electron. With the relation m/e = 2/jie/3a and /jl = 1 and
e=10~ 21 in the electromagnetic system of units, we then
have a = 10~ 14 nearly, while the radii of molecules are of the
order 10~ 8 cm., so that the linear dimensions of an electron
are about the millionth part of those of molecules. We
must therefore concede to the electron great freedom of
motion in the interstices between the molecules even of solid
bodies.
A very remarkable fact about the equations of motion of
the cathode stream used by Thomson is that, although the
velocity attained is about onethird that of light, there is no
sign of any necessity to take account of appreciable frictional
resistance. The electrons stream through the aether with
nearly the velocity of light and yet provoke no noticeable
resistance. What wonder, then, that any sethereal resistance
to planetary motion has remained beyond our ken !
The importance of the quantitative results in these experi
ments has necessitated their being discussed out of their
historical and logical order in a train of thought on cathode
and allied rays, which order we will now attempt to follow
briefly.
Stoney's interpretation of Faraday's law of Electrolysis to
mean that electricity exists in separate natural units, the
electrons, as definitely as matter in atoms, is now generally
accepted, after Helmholtz's independent advocacy of it in
his Faraday lecture.
Many workers have investigated the general dynamics of
electrons, but mostly on the supposition that the electron
must be associated with an atom, so that they form in con
junction an ion. But if electric action in matter is to be
explained only by the participation of electrons, it naturally
follows that we should contemplate the existence of electrons
in the sether to enable it to play its part in electrical action.
And next we have to take account of the hypothesis
advanced by Helmholtz in his Faraday lecture (Chem. Soc.
Trans, xxxix. 1881) to explain Contact Electromotive Force,
namely, that different atoms attract electrons with different
amounts of force. This hypothesis may not be generally
accepted yet, but we propose to follow out its logical con
sequences. If two things attract one another they must be
entities of somewdiat the same sort, and therefore the electron is
Cathode j Lenard, and Rontyen Rays. 273
of essentially the same nature as an atom. But further, if two
things attract one another, we must conceive the possibility
of their being drawn apart, so that the ion can be split into
an uncharged atom and an electron free of attachment to
matter. Maxwell's ascription of inertia to electricity, in
his theory of induced currents, bears out our conclusion that
the atom and the electron are things of the same sort in many
respects. If the electrons are distributed through the aether,
we must suppose that in aether showing no electric charge
each negative electron is united with a positive electron to
form the analogue of a material molecule, which might con
veniently be called a neutron. Of the existence of neutrons
in the aether we have powerful evidence in Trowbridge's
wonderful experiments ( u The Electrical Conductivity of the
.Ether," Phil. Mag. [5] xliii., May 1897). He opens his
account of them with a mention of Edlund's old contention
that the aether is a conductor and J. J. Thomson's refutation
of it, and closes it with the statement, " My experiments lead
me to conclude that under very high electrical stress the
aether breaks down and becomes a good conductor." Thus both
Edlund's contention and J. J. Thomson's are happily recon
ciled ; the aether is a perfect insulator until it is broken down,
after which it is a conductor. According to the present
theory, Trowbridge's result would be worded thus : — The
aether insulates until the electric force at some point is
sufficient to decompose the neutrons into electrons, where
upon it becomes a conductor of the same type as electrolytes.
This principle should help practical electricians to construct
a consistent theory of the hitherto rather intractable electric
arc.
But to return to the cathode rays. The volume of experi
mental and theoretical work on the ionization of gases, which
has been turned out from the Cavendish Laboratory, leaves no
doubt as to the existence of ions in rare gases through which
a current of electricity is passing : hence in the cathode stream
there must be a certain number of ions flying along side by
side with the electrons; but the experiments of Thomson and
Kaufmann, according to our interpretation, prove that the
stream of ions is of quite subsidiary importance to the stream
of electrons. This is not always necessarily the case in the
electric discharge through gases, and it seems to me that, for
a satisfactory theory of the varied phenomena of electric
conduction through gases, we must take account of the fact
that we have two conducting media participating in the action
namely, varying numbers of ions and also of free electrons.
Our theory of the cathode stream has the advantage that it
274 Mr. W. Sutherland on
leads in a most natural manner to a theory of the Lenard rays.
The cathode stream of electrons, moving with a velocity
nearly that of light, possessing inertia, and yet of a size that
is small compared to the molecular interspaces in solids, must
be able to penetrate a solid that is thin enough, and to emerge
on the other side, differing from the original cathode stream
only in that the small trace of moving ions has been filtered
out. Practically then Lenard rays are cathode rays. This
is what experiment has abundantly proved. All the main
properties of the cathode rays have been reobserved in the
Lenard rays: thus Perrin proves that the cathode stream
carries negative electricity, McClelland proves the same for
the Lenard rays : Eontgen discovers that where the cathode
stream strikes a solid it emits Rontgen rays ; Des Coudres
proves that where the Lenard rays strike a solid they also
emit Rontgen rays : Goldstein discovered that the cathode
stream colours salts, especially haloid salts of the alkalis, in a
remarkable way; Des Coudres proves the same for the
Lenard stream: and so on with such properties as magnetic and
electric deflectability, power of exciting luminescence, and
the like. The cathode and Lenard streams are the simplest
forms of electric current known to us. Such a power as that
of causing certain substances to emit light is only another form
of our fundamental principle, that an electron in having its
motion arrested imparts energy to the arresting molecules,
and of course to their associated electrons. The colouring of
salts discovered by Goldstein would be accounted for by the
supposition that some of the negative electrons attach them
selves to the electronegative atoms, thereby converting them
into free ions, and liberating uncharged atoms of the metal,
which cause the coloration. The experiments which have
been made, with negative results, to detect the metal or the
ion chemically do not decide anything, because of course the
amounts produced are too small for ordinary methods of
analysis to detect. The fatigue, which some substances show
after fluorescing for a while under the influence of the cathode
stream, may be' accounted for in a similar manner by the
lodgement of free electrons, which produce an opposing
electromotive force and diminish the intensity of the cathode
stream, while at the same time producing an analogous change
to the change of colour in the salts studied by Goldstein,
except that the change does not appear as visible colour, but
as a lowering of fluorescent power. Fluorescence is known
to be very sensitive to the presence of small traces of sub
stances.
We do not know enough of the relations of atoms and
Cathode, Lenard, and Rontgen Rays. 275
electrons to formulate a priori what ought to be the law of
the resistance of bodies to the passage of a stream of electrons
through them ; but fortunately we have the comprehensive
investigations of Lenard on the subject and can give a
reasonable explanation of his results. He found (Wied. Ann.
lvi.) that for a great variety of substances of densities varying
from that of hydrogen at 3 mm. of mercury pressure ('0 6 368)
to that of gold (19*3), the resistance to the passage of Lenard
rays depended almost solely on density, the coefficient of
absorption being proportional to the density. Now we should
expect our electron being so small compared to atoms, and
moving with high velocities, to deform locally any atom
which it strikes, and to rebound before the deformation had
travelled far into the substance of the atom, so that after the
electron had departed the atom would be left with an increase
of vibrational energy, but no direct appreciable increase of
translatory energy ; then, if the velocity of propagation of a
disturbance in all atoms is the same, and also the time of an
encounter between atom and electron constant, the energy
given up by an electron in an encounter with an atom will
be proportional to the density of the substance of the atom.
Now in the case of a solid, as an electron threads its way
through the molecular interspaces, the number of its encounters
will be proportional to the length of path, and therefore to
the thickness of the solid, and therefore the coefficient of
absorption, which will relate to unit thickness of all substances,
will be proportional to the density of the substance of the
atom, which is nearly the same as the density of the sub
stance ; thus for solids we interpret Lenard's law of the
absorption of cathode rays.
In the case of gases an interesting difference presents
itself. The electron is not now threading its way through
narrow passages, but has far more clear space than obstacle
ahead of it. As the electron is very small itself, we may say
that in passing through a gas the number of times it en
counters a molecule is proportional to the mean sectional area,
and therefore to the square of the radius R of the molecule
regarded as a sphere, and also to the number of molecules
per unit volume (n) ; and if m is the mass of the molecule the
density of its substance is proportional to ?n/R 3 , and thus the
coefficient of absorption for a gas is proportional to nWrn/W
or rnn/R; but nm is the density p, so that the coefficient of
absorption of a gas is proportional to the density, but also
inversely proportional to the molecular radius. Now this
theoretical conclusion corresponds partly with one of Lenard's
experimental results, namely, that although the coefficient of
276 Mr. W. Sutherland on
absorption for a large number of gases appeared to be pro
portional to the density within the limits of experimental
error, the coefficient for hydrogen was exceptional to an
extent decidedly beyond possible experimental error. In his
experiments, Lenard showed that if J is the intensity of a
Lenard stream at its source, J that at a distance r from the
source in a substance whose coefficient of absorption is A,
J=J « A 7^,
and determined A for various gases at a pressure of one atmo.
As the densities of these gases are as their molecular weights,
with that of hydrogen = 2, he shows the relation of A to the
density of different gases by tabulating values of A/m ; while
according to our reasoning RA/m would be expected to
be constant. The following table contains Lenard's values
of 10 3 A/m, and relative values of R as given in my paper on
the " Attraction of Unlike Molecules — The Diffusion of Gases,"
Phil. Mag. [5] xxxviii., being half the cuberoot of the
limiting space occupied by a grammemolecule of the sub
stance and controlled by comparison with molecular dimen
sions as given by experiments on the viscosity of gases ; the
last row contains the product 10 3 RA/m : —
H 2 . CH 4 . CO. C 2 H 4 . N 2 . 2 . C0 2 . N 2 0. SO a .
10 3 A/m ... 237 124 122 132 113 126 115 102 133
R , 1025 147 135 175 1415 134 156 1535 163
10 3 RA/w... 243 182 165 231 160 169 179 157 217
Thus while Lenard's approximate constant ranges from 102
to 237, the one to which we have been led ranges from 157 to
243, which is an improvement. The really striking point
about Lenard's discovery, however, is that when A is divided
by density, the range in value is from 2070 for paper to 5610
for hydrogen at one atmo ; the results for many substances
such as gold and hydrogen at 1/228 atmo falling between
these extremes. The fact that the value of Afp for a rare
gas is almost the same as for a dense solid, would seem to
indicate that it is only when an electron strikes an atom
almost in the direction of a normal that the most important
part of the absorption of energy occurs; for if this is so,
the chance of an electron's encountering an atom in a solid
normally, while threading its way through the interstices,
being the same as if it could pass through all the atoms which
it does not meet normally, the absorption of energy from an
electron by a number of atoms should be the same whether
they are as close as in a solid or as wide apart as in a rarefied
gas. Thus probably the coefficient of absorption for a solid
Cathode^ Lenard, and Rbntgen Rays. 217
depends on its molecular radius, but the data hardly permit
of an examination into this point. It should be remarked
that, as the electron can probably pass easily between the
atoms in a molecule, the absorption due to a compound mole
cule ought to be analysed into the parts due to its atoms ;
for instance, in Lenard's table of values for A in such gases
as CH 4 , C0 2 , C 2 H 4 , the part due to the carbon, the hydrogen,
and the oxygen, ought to be separated out, and then each
part ought to be proportional to the atomic mass and inversely
proportional to the atomic radius. If this is so, then the
agreement in the values of RA/m in our table could not be
expected to be perfect.
A very characteristic property of the Lennrd rays follows
from our theory, for when the cathode rays fall on an alu
minium window, such as Lenard used, they have a direction
normal to it, whereas the Lenard rays issuing on the other
side of the window are uniformly radiated in all directions ;
and this is exactly how our stream of small electrons would
behave, because after they have threaded their way through
the molecular interstices, they will issue with directions
uniformly distributed in space, for it is to be presumed that
the final directions of the intermolecular passages will be
distributed at random.
As the cathode and Lenard streams are currents of electrons,
and therefore form pure electric currents, we might expect
a priori that the coefficient of absorption of substances for
them would show some decided relation to the electrical
resistance of the substances ; but Lenard's law proves that
such an expectation would be futile, for the absorption of
conductor and insulator alike depends almost entirely on
density. This fact throws considerable light on the nature
of metallic conduction. It would seem as if in the conduction
of electricity in metals, both the positive and negative elec
trons, distributed through the metals, take part in the process
of conduction, probably in the method of the Grothuss chain;
by a process of exchange of partners both kinds of electron
get passed along in opposite directions without anything of
the nature of a great rush of one kind of electron at one time
and place. When such a rush occurs in a cathode stream,
the internal appliances of the best conducting metal can no
more facilitate its passage, than can the obstructing appliances
of the best insulator hinder it. In metallic conduction we
have to do with a property of the metallic atom, whereby,
with the aid of electromotive force, the local dissociation of
the neutron into electrons is greatly facilitated ; whereas in
insulators the reverse is the case. This important field of the
278 Mr. W. Sutherland on
relations of electrons and atoms must be nearly ready for
important developments.
Two more of Lenard's facts are of special importance,
namely, that cathode rays, when passed through a window
from the vacuumtube in which they are generated, travel as
Lenard rays through gas of such a density as would prevent
the formation of the cathode rays, if it prevailed in the tube,
whether that density is great, as in the ordinary atmosphere,
or very small, as in a vacuum so high as to insulate under the
electric forces in the tube. These facts are explained by our
theory : the properly exhausted tube furnishes a requisite
facility for splitting up the neutrons and getting a supply of
electrons to be set swiftly in motion ; once that is accomplished,
nothing will stop them until it offers enough resistance to
destroy the momentum of the electrons, and ordinary lengths
of dense or rare air in Lenard's experiments failed to do this.
The action of the tube in generating the cathode rays may be
likened in this connexion to a Gifford's Steam Injector.
In the logical development of the present line of thought,
an attempt at an explanation of the cause of the Rontgen
rays must find a place. Already J. J. Thomson, in his paper
on a Connexion between Cathode and Rontgen Rays (Phil.
Mag. [5] xlv., Feb. 1898), has worked out in some detail the
electromagnetic effect of suddenly stopping ions moving with
high velocity, the main result being that thin electromagnetic
pulses radiate from the ion. He believes that these pulses
constitute the Rontgen rays, in agreement with a surmise of
Stokes. Thomson's reasoning would apply to our free
electrons just as to his ions, but there would be this important
distinction, that while Thomson's hypothesis involves the
condition that the greater part of the energy of the cathode
stream consists of the kinetic energy of the atoms, in our
hypothesis the energy belongs almost entirely to the moving
electrons, and when these are stopped the energy appears as
heat at the place of stoppage. Thus Thomson's electromag
netic pulses appear only as subsidiary phenomena in con
nexion with the conversion of the kinetic energy of the
electrons into heat ; indeed, we cannot be sure that they
exist, because their existence has been suggested only in
accordance with the particular assumptions in Thomson's
hypothesis which correspond to only a limited portion of the
complete electrodynamics of such an action as is contemplated
in this paper, causing the conversion of all or almost all the
kinetic energy of an electron into heat. Moreover, in tracing
the relation of Lenard rays to cathode rays we have been led
to picture the stoppage of the moving electrons as nothing
Cathode , Lenard, and Rontgen Rays. 279
like so sudden as that which Thomson has to contemplate for
his charges : this of course only makes a difference in the
degree of intensity of the phenomena resulting from the
stoppage. There has as yet been no systematic proof that
the properties of a train of impulses would be the same as
those of the Rontgen rays in the matter of the absence of
refraction and reflexion. Again, it is recognized that the
Rontgen rays and the Becquerel rays from uranium are very
similar, but it would be hard to imagine the Becquerel rays
to be due to thin impulses. On these grounds it seems to
me that Thomson's suggestion as to the cause of the Rontgen
rays, although exciting one's admiration by its clear con
sistency, does not lead to the desired end ; and therefore I
will try to follow out the premisses of this paper to such
conclusions as may relate to phenomena like those of the
Rontgen rays.
To the electrons we have assigned inertia and size, and we
must therefore ascribe to them shape ; but a general con
ception of shape involves also the notion of deformability,
which, therefore, we must consider as a possible property of
the electron. The electron is therefore to be supposed
capable of emitting vibrations due to the relative motions of
its parts ; as light is supposed to be due to the motion of
electrons as wholes, we see that the internal vibrations
of electrons will have this much in common with light, that
they are transmitted by the same sether, but they need have
nothing else in common. We propose, then, to identify the
Rontgen rays with these internal vibrations of our electrons.
It might be expected that the electron, in executing the
motions which cause light, would get strained and thrown
into internal vibration, so that Rontgen rays would accom
pany ordinary light ; but the fact that Rontgen rays cannot
be detected in association with light shows that the motion
of the electron occurs either so that it is very free from shock
and strain, or so that atoms promptly damp any internal
vibrations of adjacent electrons. The way in which matter
absorbs the energy of Rontgen rays shows that we may rely
on atoms to suppress any small amount of Rontgen radiation
that might tend to accompany ordinary light as emitted by
electrons. Thus, then, appreciable Rontgen radiation is to be
looked for only when free electrons are thrown into vigorous
internal vibration. Now the encounter of an electron with
an atom, in which it gives up a part of its kinetic energy to
the atom as heat, is precisely the sort of action by which we
should expect the electron to be thrown into internal vibration.
Internal vibrations should originate where cathode or Lenard
2 80 Mr. W. Sutherland on
rays are absorbed, and most powerfully where the absorption
is most powerful : this corresponds with all the facts as to
the place of origin of Ronigen rays.
As to what must be the order of magnitude of the length
of the waves in the aether produced by the internal vibrations
of the electrons, we can form no a priori estimate, but under
the circumstances we are at liberty to assume that, like the
size of the electron, it is small compared to that of atoms, and
small also compared to molecular interspaces. We shall
then have to do with systems of waves, which, when they
fall on a body, can travel freely in the molecular interspaces,
but are liable to absorption near the surfaces of molecules.
The propagation of such a system of waves would take place
almost entirely in the a?ther of the interspaces, as sound travels
through a loose pile of stones mostly by the airspaces ; the
molecules cause absorption, but do not act as if they loaded
the sether. Therefore when our system of waves enters a
body it experiences no refraction. As to reflexion at the first
layer of molecules which it encounters, we must remember
that our wavelength is small compared to the radius of a
molecule or atom, and that therefore in studying reflexion it
suffices to study that from a single molecule ; whereas, with
ordinary light, where the wavelength is large compared to
atomic radius, we have to take the effect of a large number
of contiguous molecules, if we are to reason out results com
parable with those observed in ordinary reflexion. Now the
reflexion of our small waves from a single molecule will be of
the same nature as reflexion from a sphere, and will be similar
to diffuse scattering, a good deal of the scattering being
towards the neighbouring interstices. Thus the attempt to
reflect these waves from a material plane surface will be
similar to that of attempting to reflect ordinary light from a
large number of smooth spheres whose centres lie in a plane.
If we take the average effect of a large number of molecules
whose centres are by no means in a plane, as must be the
case with our best reflecting surfaces, we see that a diffuse
scattering of our small waves must take the place of re
flexion, and this is the experimental result with the Rontgen
rays.
Any polarization that our system of waves might possess
could not be detected by the ordinar}' optical appliances,
because these depend on actions exercised by the molecules
on the vibrations of light, whereas, as our small waves travel
by the interstices between the molecules, their character is
not controlled to any appreciable extent by the molecular
structure. This result also agrees with the experimental one
Cathode, Lenard, and Rontgen Rays, 281
that polarization of the Rontgen rays cannot be detected by
ordinary optical apparatus for the purpose.
These negative properties have been explained chiefly by
the assumed smallness of the wavelength, and have, there
fore, little direct connexion with our theory of the Rontgen
rays beyond indicating the probability of a small wavelength
for the Rontgen rays for similar reasons to those usually
urged. We must, therefore, proceed to properties that our
short waves must possess by virtue of their origin in vibrating
electrons. In the first place, we should expect the electrons
forming the neutrons in the aether to be set vibrating by our
waves ; but if they produce no dissipation of the energy they
will not cause any absorption, but will simply participate in the
general asthereal operations of propagating the waves. But
when the waves get amongst the electrons associated with
atoms, and set them vibrating internally, there is called forth
that resistance to the vibration which constitutes the damping
action already spoken of. One of the probable results of such
an action would be the setting of the acting and reacting
atom and electron into relative motion, so causing the ab
sorbed Rontgen energy to appear as some form of radiant
energy congenial to the atom and electron. In this way our
waves could give rise to fluorescent and photographic effects
in the manner of the Rontgen rays. If an electron absorbs
enough of the energy of our small waves, it may be set into
such vigorous motion as to escape from the atom with which
it is acting and reacting, and appear as a free electron, or it
may associate itself with an electron* to form an ion. At
the foundation of our theory we suppose our small waves to
be produced by the deformation of an electron during a
vigorous transfer of energy from electron to atom ; and now
we suppose this to be a reversible action, so that an electron
set vibrating near to an atom can convert enough of its
vibrational energy into translational kinetic energy to escape
from the atom. With this legitimate dynamic assumption of
reversibility, we can deduce from our hypothesis the produc
tion of free electrons or ions in a dielectric traversed by our
small waves, which is in agreement with the remarkable pro
perty possessed by the Rontgen rays of making gases con
duct electricity well. The presence of scattered ions in a
solid dielectric does not necessarily make it conduct. An
experimental method of testing our theoretical conclusion,
that Rontgen rays ought to have the same effect on solid
dielectrics as on gases, would be to heat one till it gave de
cided signs of electrolytic conduction, and then test as to
* [Atom ?]
282 Mr. W. Sutherland on
whether conductivity is increased by radiation with Rontgen
rays. Experiments on liquid dielectrics should be easy enough.
One of Rontgen's observations is of special importance. He
found that air, through which Rontgen rays are passing,
emits Rontgen rays ; and this is exactly what our theory
would indicate, because, as we saw in discussing reflexion,
each atom scatters our small waves as a reflecting sphere dis
tributes ordinary light.
The remaining important positive facts concerning Rontgen
rays relate to their absorption in passing through different
substances. Our short waves in passing through a unit cube
of substance in a direction parallel to one of the edges, while
passing along the molecular interspaces, will be falling at
intervals directly on opposing surfaces of atoms; and if n be
the number of atoms per unit volume, and It the radius of
each, the quantity of surface encountered by unit area of
wavefront will be proportional to ?itR 2 , and the number
of encounters in passing unit distance will be proportional to
ni, so that as regards amount of encounter of wavefront with
atoms the energy absorbed by the atoms will be proportional
to 7?R 2 . But if the effectiveness of a collision in causing ab
sorption from a given area of wavefront in a given time is
also proportional to the density of the matter in the atom,
that is to m/R 3 , as we had to suppose in discussing the col
lisions of electrons and atoms, then the absorption of energy
from our short waves in passage through unit length of
different substances will be proportional to n?///R, that is to
the density and inversely to atomic radius as with Lenard
rays. The fact that Rontgen rays produce powerful fluores
cence in certain substances shows that there are special re
sonance phenomena that must be expected to produce decided
variations in absorption from the simple form just discussed ;
but the fact remains, as discovered by Rontgen, that by far
the most important factor in the absorption of Rontgen rays
is density. Benoist (Compt. Rend, cxxiv.) has found that
the absorption of Rontgen rays by certain gases is propor
tional to the density, the factor of proportionality being
nearly the same as for solids such as mica, phosphorus, and
aluminium, though rising to a value six times as great in the
case of platinum and palladium ; density is the prevailing,
but not the only, property which determines the absorption
of the Rontgen rays. But under different circumstances
Rontgenray apparatus gives out rays of very different absorb
ability, or, as it is usually expressed, of different penetrative
power. Thomson's theory of the Rontgen rays, as thin elec
tromagnetic pulses, does not seem to offer any feasible ex
Cathode, Lenard, and Rontgen Rays. 283
plana tion of this fundamental fact. The theory of vibrating
electrons requires that, in addition to the fundamental mode
of vibration, we must contemplate a number of harmonics
associated with it ; various combinations of fundamental and
harmonics will be associated with different conditions of
generation of the vibrations, and these will correspond to the
Rontgen rays of different penetrative power.
An interesting observation of Swinton's, that two colliding
cathode streams do not give rise to Rontgen rays, is explained
by our hypothesis, because the electrons are so small and so
far apart that an appreciable number of collisions between
the electrons of two colliding streams cannot occur.
Some consequences of our line of reasoning, to which as
yet no corresponding experimental results have been obtained,
may now be indicated. The difference between cathode and
anode is due to the fact that the attraction of metallic atoms
for positive electrons is stronger than for negative ones, so
that under a given electrical stress negative electrons break
away as a cathode stream more easily than positive ones as
an anode stream. But still, under strong enough electric
stress at the anode, it ought to be possible to get an anode
stream or anode rays similar to the cathode rays, but carry
ing positive electricity. These on encountering atoms, espe
cially the atoms of a solid body, should cause the emission of
rays similar to the Rontgen, but possibly very different in
detailed properties, such as wavelength. It is possible that
the Becquerel rays may be examples of what we may call
positive Rontgen rays, because, while we have seen that, in
the majority of cases, electrons move relatively to atoms in
the production of light, in such a manner that they do not
experience shocks throwing them into internal vibration, the
uranium atom may be so formed that it periodically collides
with its satellite electron or electrons, in which case the atoms
of uranium would be a source of radiation analogous to the
Rontgen.
According to our theory the velocity of the cathode stream
is not a physical constant like the velocity of light through
the aether, but ought to vary greatly according to the history
of the stream, which starts with zero velocity and ends with
the same. The velocity of the Rontgen rays should be of
the order of that of light : we cannot assert that it should be
exactly equal to that of light, because to waves of so short a
length the neutrons may act as if they loaded the aether, so
that Rontgen rays may suffer a refraction in aether in com
parison with light. The fact that the experimental velocities
found for the cathode rays are of the order of the velocity of
284 Dr. R. A. Lehfeldt on the
light is a striking one, to be compared with the fact that in
the vena contracta of a gas escaping from an orifice the
maximum velocity attainable is nearly that of the agitation
of the average molecule in the containing vessel or of sound
in the gas.
It appears as though a complete theory of electricity would
be a kinetic theory, in which the place of the atoms or mole
cules of the kinetic theory of matter is taken by the electrons.
The ion appears as a sort of molecule formed by the union of
an atom or radical to an electron. But such large questions
can hardly be opened up in the present connexion. We may
summarize the contentions of the preceding pages in the two
propositions : —
The cathode and Lenard rays are streams, not of ions, but
of free negative electrons.
The Rontgen rays are caused by the internal vibrations of
free electrons.
Melbourne, Nov. 1898.
XXI. Properties of Liquid Mixtures. — Part III.* Partially
Miscible Liquids. By R. A. Lehfeldt, D.Sc]
THE phenomena of complete mixture between two liquids,
about which so little systematic knowledge is yet in
existence, are connected with the phenomena of ordinary
solution by an intermediate stage, that in which two liquids
dissolve one another to a limited extent only. The study of
such couples seems a promising field of investigation, on
account of the intermediate position they occupy; it seems
to offer the chance of extending some of the laws arrived at
with regard to simple solution to the more complicated cases;
I have therefore attempted to get some information on the
equilibrium between incomplete mixtures and the vapour
over them, and especially at the " critical point," i. e., the
point at which incomplete miscibility passes over into com
plete. A recent short paper by Ostwald % draws attention
to the importance of that point in the theory of mixtures.
Choice of Liquids.
The first point is to obtain suitable pairs of liquids for
experiment. In order to study the properties of the critical
point with ordinary vapourpressure apparatus, it is necessary
* Part I. Phil. Mag. (5) xl. p. 398; Part II. Phil. Mag. (5) xlvi.
p. 46 ; reprinted, Proc. Phys. Soc. xvi. p. 83.
t Communicated by the Physical Society : read Nov. 25, 1898.
X Wied. Ann. lxiii. p. 336 (1897).
Properties of Liquid Mixtures. 285
that the pressure at the critical point should be below one
atmosphere, and that limits very much the choice of liquids.
As a rule, two liquids either mix completely cold, or if they
do not do that, raising to the boilingpoint does not suffice to
make them mix ; two or three cases are all that I have been
able to rind in which the point of complete mixture can be
arrived at by boiling, and consequently corresponds to a
vapourpressure below the atmospheric. Many other pairs
of incompletely miscible liquids have been studied by
Alexejew and others, but to arrive at their critical points it
was necessary to raise them to a high temperature in sealed
tubes. A recent paper by Victor Rothmund * contains new
observations on the relation between concentration and tem
perature, including the concentration and temperature of
the critical point, made by Alexejew's method. That paper
contains a long account of previous work on the subject,
which makes it the less necessary for me to go over the same
ground. I will therefore mention only what has been done
on vapourpressures, as Rothmund does not touch on that
side of the subject, merely adding two remarks to his paper.
First, it does not seem to have been noticed that normal
organic liquids always mix completely : I hoped to have
found a normal pair to study first, in order to avoid the com
plication due to the abnormality supposed to be molecular
aooregation in the liquid ; I have not succeeded in finding
such a pair. All the incompletely miscible pairs of liquids
so far noted include water, methyl alcohol, or a low fatty
acid as one member. To those with accessible critical
points mentioned by Rothmund, I have only one pair to add,
viz., ethylene dibromide and formic acid ; these mix on
boiling and separate into two layers when cold. I have not
yet gone further with this couple ; the vapourpressure
observations below refer to the wellknown cases of phenol
and water, aniline and water.
An account of previous experiments on the vapourpressure
of incompletely miscible liquids will indeed not take up much
space, since, so far as I know, there is only one to record,
viz., Konowalow's f measurements on isobutyl alcohol water
mixtures. His observations (made by a static method) give
some points on the vapourpressure curve up to 100° for
(1) pure isobutyl alcohol (100 °/ ) : (2) mixtures containing
94*05 °/o an d 6*17 °/o>both clear; (3) an undetermined mixture
which separated into two layers. He unfortunately did not
measure the solubility of the alcohol in water, or water in
* Zeitschr.f.phys. Chem. xxvi. p. 433 (1898).
f Wied. Ann. xiv. p. 43.
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. X
286 Dr. R. A. Lehfeldt on the
the alcohol, at any of the temperatures for which vapour
pressure observations are recorded, so those data have to be
supplied from Alexejew's results *. Konowalow, in the
second part  of his paper, proceeds to show that the possible
forms of curve showing vapourpressure against concentration
(temperature constant) are two : (i.) the flat part of the
curve bounded by a rising portion at one end and a falling
portion at the other ; (ii.) the flat part bounded by a falling
portion at each end. Isobutyl alcohol water mixtures give a
curve of the latter kind.
Isobutyl alcohol and water, however, possess a critical
point at about 130°, i. e., much above the boilingpoint of
either. I therefore decided to study first mixtures of phenol
and water, which become homogeneous in any proportions
below 70°.
The phenol was a commercial " pure " specimen ; to purify
it further, it was placed in a distillationflask and melted ;
then air was drawn through it for about half an hour, whilst
its temperature was kept at about 160° to 170°, in order to
dry it. It w T as then distilled, and by far the larger part came
over between 178° and 180°. The fraction collected between
179 0, 5 and 180° (about half the mass) was used in the
experiments. To make up mixtures, the process always
adopted was to warm the stoppered bottle containing the
phenol to just above the meltingpoint, and pour the required
amount into a wei ohingflask. It was found that the moisture
absorbed from the air during the process was quite inappre
ciable. The phenol, kept day after day at 40° to 50° ready
for use, slowly turned pink, showing the presence of rosolic
acid ; but a comparative colourobservation showed that the
amount of impurity was probablv not more than 1/10,000.
When it was necessary to estimate phenol in a mixture, that
was done by the method of Koppeschaar, tribromphenol
being formed and the excess of bromine replaced by iodine
and titrated with thiosulphate. The method gave quite
satisfactory results.
Experimental Methods used.
The measurements on phenolmixtures gave results con
trary to my expectations, so that I became suspicious of the
experimental methods. In the end I made use of four
different kinds of apparatus, but found that they gave results
in practical agreement, so that it became chiefly a question of
convenience to decide between them.
* Wied. Ann. xxviii. p. 315. f Wied. Ann. xiv. p. 222.
Properties of Liquid Mixtures. 287
The first method tried was the " dynamic," carried out
with the same apparatus as described in Part II. It required
no modification, except the use of a new thermometer, since
the old one did not go above 60°. The new thermometer was
a longer one, graduated in ^ from 0° to 100° (by C. E.
Midler, ^No. 8). Its corrections were obtained in two ways :
first by comparison with a standard (Reichsanstalt, 7347) at
certain fixed temperatures, viz., the boilingpoints of methyl
acetate (57°), methyl alcohol (65°), and ethyl alcohol (78°);
secondly, by measurements of the vapourpressures of water
under the same conditions as in the actual experiments ; in
these conditions part of the stem w r as exposed.
To use the apparatus the required mixture was weighed out
from melted phenol and distilled water, then warmed up in
the w T eighingbottle until it became homogeneous, and poured
into the tube of the vapourpressure apparatus. The apparatus
works satisfactorily except for mixtures on a very steep part
of the curve of vapourpressure (p) over concentration (z) ;
w T hen dpjdz is great, the change of composition of the liquid,
due to the evaporation becomes disproportionately important,
and the static method is to be preferred ; in the case of phenol
mixtures, however, that only affects a small part of the range
— mixtures with 90 per cent, or more of phenol.
The anomalous result that made me at first doubtful of the
accuracy of the method was that up to 60 or 70 per cent, of
phenol added to water made practically no difference to the
vapourpressure of the water. To check this, I made one or
two experiments by the static method, in a barometer tube
surrounded by an alcoholvapour jacket of the usual pattern.
They were not carried out with any attempt at accuracy, but
sufficed to show that the previous observations could not be
far wrong. The problem then was to determine the small
difference in pressure between water and the phenol water
mixtures, and as for that purpose a differential gauge is
obviously more appropriate, I set about designing and making
the apparatus described below. Its design is based on a point
of technique that does not seem to be much known, and to
which, therefore, I should like to draw attention. If a glass
tube be drawn out fine, sealed at one end, and evacuated, the
sealed end may be broken under the surface of a liquid,
which then flows in at any desired rate according to the
diameter of the tube, and the tube may at any moment be
fused off in the middle by a mouth blowpipe, without any
inconvenience whatever. This process of filling with a liquid
will I think be found advantageous in many cases. The only
trouble about it is to get the canillary of the right bore";
X2
288 Dr. R. A. Lehfeldt on the
since the rate of flow depends on the fourth power of the
radius it is easy to make the tube too wide or too narrow.
Of course I made a good many failures at first, but after
some practice could rely on getting the required condition.
I used tubing of about 1 millim. internal diameter, and
4 millim. external, and drew it out till the internal diameter
was about one sixth of a millimetre ; a few centimetres of
such a bore gives a convenient rate of flow for liquids of the
viscosity of water.
Ficr. 1.
AT
D'
The apparatus for vapourpressure measurements is shown
in fig. 1. It consists of a Utube, A, to serve as a gauge,
carrying a branch, B, below, drawn out for filling as mentioned
above. The top of the gaugetube is bent round each side to
the bulbs C, C, which are also provided with filling tubes
Properties of Liquid Mixtures. 289
D, D'. The whole is shown flat in the diagram ; but as a
matter of fact the side tubes C D and C D' were bent round
till the bulbs nearly touched, to ensure their being of the same
temperature. The apparatus was cleaned out with chromic
acid, washed, and dried ; the capillaries were then drawn out
and two of them sealed up, the third being left with the bit
of wider tubing beyond the capillary untouched. By means
of this it was attached to a mercurypump, exhausted, and
the capillary fused. The point B was then opened under
mercury and fused off when the gauge contained sufficient :
in the same way one of the bulbs was half filled with the
mixture through D, arid then the other with water (which
must, of course, be freed from air) through D' '. The apparatus,
all of glass and hermetically sealed, is then ready for use : a
glass millimetre scale is fastened with rubber bands to the
gaugetube, and it was immersed in a large glass jar of water.
The scale was usually read by the telescope of a cathetometer
and sometimes the screw micrometer of the telescope used
to subdivide the graduation. The differential method avoids
the necessity for any very great care in maintaining or mea
suring the temperature of the apparatus. It was found quite
sufficient to heat the waterbath by leading a current of
steam into it, and when the required temperature was
reached, stop the steam for a moment and read the differ
ence of level. When the highest temperature (90°) was
reached, some of the water was siphoned off, replaced by
cold, the whole mass well stirred, and a reading taken. There
was no noticeable lag in the indications of the gauge, the
readings at the same temperature, rising and falling, being
in good agreement.
The fourth apparatus used was the Beckmann boilingpoint
apparatus, in its usual (second) form : with that observations
at 100° were obtained of a kind to confirm the measurements
made at somewhat lower temperatures with the vapour?
pressure apparatus.
Observations of Vapourpressure.
The following observations were obtained with the differ?
ential pressuregauge : t is the temper,: turo centigrade, p
the vapourpressure of the mixture, tt that of water, tt— p
is therefore the difference observed with the gauge, and
{it— p)/7r represents the relative lowering of the vapour
pressure of water by the addition of the quantity of phenol
mentioned.
290 Dr. R. A. Lehfeldt on the
Phenolwater Mixtures.
67*36 per cent, of phenol.
t 50° 65° 75° 85° 90°
Tvp 05 04 22 56 82
(tt^/tt 0005 0002 0008 0013 0015
77*82 per cent, of phenol.
t 70° 75° 80° 85° 90°
7Tp 26 50 82 125 171
(tt^/tt 0011 0017 0023 0029 0032
82*70 per cent, of phenol.
t 40° 50° 60° 65° 70° 75° 80° 85° 90°
7Tp < 14 56 111 148 192 252 322 423 539
(jrjp)/*.. 0025 0061 0075 0079 0082 0087 0091 0098 0103
90*46 per cent, of phenol.
t 25° 40° 50° 60° 65° 70° 75°
ttp 19 91 175 317 432 535 68 6
(tt^V'tt... 0081 0165 0190 0213 0231 0230 0237
A mixture containing 7*74 per cent, of phenol gave no
certain indication of a difference of pressure between the
mixture and pure water. On this point, however, more
reliable information is to be obtained with the Beckmann
apparatus. It will be noticed from the preceding figures
that the influence of the dissolved phenol becomes steadily
greater as the temperature rises, e.g. 82 per cent, of phenol
produces nearly twice as great a change of vapourpressure
proportionally at 90° as it does at 50°. In agreement with
this the rise of vapourpressures in dilute solutions of phenol
is more marked at 100° than at the lower temperatures at
which the vapourpressure apparatus is available. The result
of an experiment on the boilingpoint is as follows : —
Per cent,
phenol.
Fall of
boilingpoint.
Corresponding
rise of vapour
pressure, p — 7T.
48
90
130
164
0154
0169
0161
0154
41
45
43
41
The general character of the results is sufficiently shown
Properties of Liquid Mixtures.
291
by fig. 2, in which the isothermals of 90° and 75° are suffi
ciently represented. That of 90° is comparable with the curve
I
ig.
2_
Isotherm
als
of Phenol Water Mixtures.
5U
\
\
\
\
•7 i
\
/ 3
f*s
<* — r
©■
V
v
\
\
\
+c
\
\
\
\\
, ^
_
©
^u
\\
A
X
\
\
\>
■ ■
i
509
400
300 B
o
a
>
100
mm.
50
100
Per cent, phenol in water.
for alcoholtoluene mixtures (see fig. 2 in the preceding
memoir), only that the flatness extending over a great part of
the range of concentrations is exaggerated in the phenol water
mixtures. The curve for 75° — still above the critical point —
is still flatter; indeed it is imposible to say whether it rises or
falls. Probably, therefore, below the critical point (where
the vapourpressure of phenol is inconveniently small for
measurement) the isothermal, instead of consisting of a hori
zontal line bounded by two curves, would consist of a hori
zontal passing through the point representing pure water,
bounded at the other end only by a descending curve. Such
292
Dr. R. A. Lehfeldt on the
an isothermal — that of 50° — is shown in fig. 2. The hori
zontal part ends at the point A (63 per cent.), beyond which
the mixtures are homogeneous : the curve beyond A may
theoretically meet the horizontal line at a finite angle, but
that is certainly not distinguishable on the diagram. The
curve is in fact exactly similar, so far as the experiments
show, to that for 75°, and the pressure of the critical point, C
on the diagram, which lies between them (at 68°*4) appears to
make no difference whatever in this case— a case of great
disparity in the vapourpressures of the two components.
For comparison,, a few experiments were made with aniline
(not specially purified) and water. A mixture which consisted
of two layers, even at the highest temperature used in the
experiment, gave in the differential apparatus the following
results : —
70°
68
00290
7T — p
(tt— p)U
80°
85°
111
133
00315
00307
Whilst the Beckmann apparatus
gave at 100° :—
Per cent,
aniline.
Lowering of
boilingpoint.
7Tp.
IT — p
— ■
399
0636
171
00225
768
0921
246
00324
1110
0921
246
00324
The second column gives the observed fall in temperature
on adding aniline to the water ; the third column the rise of
pressure corresponding, at the rate of 26*8 millim. per degree.
Water at 100° is saturated by the addition of 6' 5 per cent,
of aniline, and it will be seen that the vapourpressure rises
no further after that. The relative rise of vapourpressure on
saturation is 0*0324 at 100°, in satisfactory agreement with
the numbers obtained by the differential gauge (0*0290,
0*0 515, 0*0307), a tendency to increase with temperature
being distinguishable here, as with phenol. Now suppose
the vapourpressure of a saturated mixture to be obtained in
in this way : let the partial pressure of the watervapour be
that of pure water reduced by the normal amount (Raoult's
law) due to the solution in it of the maximum quantity of
aniline : and let the partial pressure of anilinevapour be that
of pure aniline reduced by the normal amount due to the
Properties of Liquid Mixtures. 293
solution in it of the maximum quantity of water. We get
the following results at 100° : — ■
Vapourpressure of water = 760 millim.
Solubility of aniline = 6*5 per cent. = 1*32 molecular
per cent.
Partial pressure of water = 98*68 per cent, of 760
= 7499 millim.
Vapourpressure of aniline = 46 millim.*
Solubility of water in aniline = 8*7 per cent. = 33 mole
cular per cent. (Alexejew).
Partial pressure of aniline = 30'8.
Total pressure = 749*9 + 30*8 = 780*7.
Observed pressure = 784*6 millim.
The vapourpressure of the saturated mixture is therefore
given fairly well by the above rule. The rule cannot be
applied to phenol mixtures, as below the critical point the
vapourpressure of phenol is too low to determine with
accuracy. Konowalow's measurements of the vapourpressure
of isobutyl alcoholwater mixtures, combined with Alexejew's
measurements of solubility, give the following results. At
90°:—
Vapourpressure of water = 525 millim.
Solubility of isobutyl alcohol = 7 per cent. = 1*8 mole
cular per cent.
Partial pressure of watervapour = 98*2 per cent, of
525 = 5155.
Vapourpressure of isobutyl alcohol = 378 millim.
Solubility of water = 25 per cent. = 57*8 molecular per
cent.
Partial pressure of isobutyl alcohol = 159*5.
Sum = 675*1.
Observed pressure =767.
In this case the alcohol saturated with water contains more
molecules of water than of alcohol, and it is not to be expected
that the normal depression of the vapourpressure should
hold over so wide a range as 57*8 per cent. The numbers
in fact show that the partial pressure of isobutyl alcohol
must be very much greater — about 250 millim. The curve
of partial pressures is therefore comparable with that for
ethyl alcohol in benzene and toluene (see Part II. tables
p. 53).
* Kahlbaura, Zeitsch. f. phys. Chem. xxvi. p. 604.
294
Dr. R. A. Lehfeldt on the
Characteristic Surface for PhenolWater Mixtures.
To complete an account of the behaviour of phenolwater
mixtures, it is necessary to draw a diagram of the relations
between temperature and concentration ; this is given in
fig. 3. Figs. 2 and 3 together, therefore, give a notion of the
shape of the " characteristic surface," i. e. the surface showing
the relations between concentration, temperature, andpressure.
Fig. 2 contains three sections at right angles to the axis of
llHi
Fie
?.s
Q
X
/
/
/
/
X
/
s
V
/
\
\
V
/
1
\
/
^
1
/
\
\
\
_H
'
*
ri;
ft!
 is
"
^
...
,"
N
F
50°
20
50
100
Per cent, phenol in water.
temperature (for T = 90°, T = 75°, T=50° respectively), while
fig. 3 gives one section at right angles to the axis of pressure
(jo = l atmo).
The behaviour of phenolwater mixtures is formally
Properties of Liquid Mixtures. 295
identical with that of benzoic acid and water*, but the curve
brnnches of the diagram are of very different relative sizes to
those of the lastnamed mixtures. The features of the diagram
are as follows : —
L. Freezingpoint of water.
0. Freezingpoint of phenol.
LS. Freezingpoint of aqueous solutions of phenol.
ONFGHMS. Freezingpoint of solutions of water in phenol.
S. Cryohydric point.
MC. Saturation of water with phenol.
NC. Saturation of (liquid) phenol with water.
C. Critical point of mixture.
The line LS is given by the thermodynamic equation
002T 2
L
where T is the absolute temperature of fusion of ice, L the
latent heat of fusion, and t the resulting molecular depression
of the freezingpoint ; it accordingly starts with a slope of
o, 2 for one per cent, of phenol. The initial slope of ON is
given by a similar equation, and is 4°'15 for one per cent, of
water ; a direct observation gave as a point of the curve
80*5 per cent, phenol,
meltingpoint + 5*0.
This is marked with a dot in the figure, and lies to the right
of ON ; by continuing the curve through the point so found
until CN is met, we reach the point N where the phenol is
saturated with water ; on increasing the concentration a
second liquid layer appears, consisting of water saturated
with phenol. NFGHM is purely hypothetical, referring to
unstable mixtures ; actually any mixture of concentration
between 8 per cent, and 77 per cent, of phenol will separate
into two layers on cooling, and on further reduction of
temperature freeze at the constant temperature (about f 1 0, 5)
represented by the horizontal straight line MN. The cryo
hydric point lies to the left of the saturation curve CM, so
that it is actually attainable : its existence was shown by
making a solution containing 5 '25 per cent, of phenol, and
cooling it in a bath of ice and salt : it began to freeze with
out previous separation into two layers, and the temperature
remained constant at — O 9. About half of it was frozen, the
beaker removed from the freezingmixture, and some of the
liquid remaining poured off for analysis ; it was found to
* See van't Hoff, Vorlesungen ihber theoretische und physikalische
Chemie, Heft i. p. 48. (Braunschweig, 1898.)
296 Mr. W. B. Morton on the Propagation of
contain 4*83 per cent, phenol : this concentration is therefore
in equilibrium with both ice and solid phenol which had been
deposited on the sides of the beaker. The cryohydric mixture
therefore contains so little phenol that it may be looked upon
as a dilute solution of phenol in water, and its calculated
freezingpoint, according to van't HofPs rule, would be — 1°'0,
in agreement with the observed value — o, 9. Consequently
solutions of strength between M and S will deposit phenol
on cooling, those between L and S (0 to 4*83 per cent.) ice.
The diagram is completed by the curve 5lCN which is
drawn from Rothmund's observations*, which are indicated
by dots ; my own observations (shown by crosses) are in
practical agreement with bis and Alexejew's.
Finally, the curves divide the diagram into regions, with
the following meanings : —
Below LS undercooled solutions of phenol, from which ice
crystallizes out, with formation of the saturated solutions LS.
Below SMNO supersaturated solutions of phenol, from
which phenol crystallizes out with formation of the saturated
solutions of phenol in water (SM) and water in phenol (NO).
MONGM, unstable mixtures which separate into the two
saturated solutions CM and CN, forming two liquid layers.
Above LSMCNO homogeneous liquid mixtures.
The DavyFaraday Research Laboratory, Royal Institution,
London, October, 1898.
XXII. On the Propagation of Damped Electrical Oscillations
along Parallel Wires. By W. B. Morton, M.A.\
IN a paper published in the Philosophical Magazine for
►September 1898 Dr. E. H. Barton has compared the
attenuation of electrical waves in their passage along parallel
wires, as experimentally determined by him, with the formula
given by Mr. Heaviside in his theory of long waves. The
results show a large discrepancy between the theory and the
experiments, the observed value of the attenuation constant
being about twice too large. Dr. Barton discusses several
possible causes of error and finds them inadequate, and
suggests that the reason of the difference may lie either in
(1) the nearness of the wires to one another, or (2) in the
damping of the wavetrain propagated by the oscillator. To
these may be added (3) the consideration that the formulae
used were deduced by Mr. Heaviside from the discussion of
his " distortionless circuit," in which the matter is simplified
* L. c. p. 452.
t Communicated by the Physical Society : read Nov. 11, 1898.
Damped Electrical Oscillations along Parallel Wires. 297
by supposing sufficient leakage to counteract the distortion
produced by the resistance of the leads, whereas in Dr. Barton's
circuit the leakage was negligible.
It is probable that the nearness of the wires has an appre
ciable effect on the phenomenon. The discrepancy would be
diminished if the actual resistance of the wires was greater
than that calculated by Dr. Barton from Lord Rayleigh's
highfrequency formula. Now the effect of the neighbourhood
of two wires carrying rapidly oscillating currents in opposite
directions is to make the currents concentrate towards the
inner sides of the wires*; and this would cause an increase
in the effective resistance.
I have examined the effects of (2) and (3), viz. of the
damping and the want of balance in the constants of the
circuit. The investigation is perhaps of some interest owing
to the fact that these elements are always present in the
ordinary experimental conditions ; although, as will be seen,
we are led to the conclusion that in all actual cases their in
fluence on the phenomena is of quite negligible order. The
method is the same as that used by Mr. Heaviside.
General Theory. — Let the inductance of the circuit be L, its
capacity 8, its resistance (of double wires) R, and its leakage
conductance K, all per unit length. An important part is
played by the ratios j and ^ ; we shall call these p and cr.
When p and a are equal we have the " distortionless " circuit
above referred to.
Now if V be the difference of potential between the wires
and C the current in the positive wire, we have the equations
S=( R+I 4) C > «
§=(*<>> w
giving
since LSt> 2 =^l, where v is the velocity of radiation.
To simplify the algebra we shall work first withV= Y Q e~ mz+ni .
* Cf. J. J. Thomson, ' Recent Researches/ p. 511.
298 Mr. W. B. Morton on the Propagation of
which can be made to represent damped periodic vibrations
by giving complex values to m and n.
The equation (3) now becomes
mV=(/D + ra)(o + 7i)s (4)
and the connexion between C and V is given by (1), viz.
L R^Ln ^
To find the effect of a pure resistance B/ between the ends
of the wires, as in Dr. Barton's experiments, put Y 1? Y 2 ,
G 1} G 2 for the potentials and currents in the incident and
reflected waves respectively. Then we have
p _ mYi n _ mVo
B + W z R+hn 3
also the total potentialdifference V x + Y 2 is connected with
total current G 1 iG 2 through resistance B' by Ohm's law,
Vj+V^B^Ci + Co).
These equations give for the reflexion factor
Vj _ B + L?z7»B /
Y x B + Lw + mB/ (6;
If the circuit be distortionless and B/ = Lv, then, as
Mr. Heaviside showed, the absorption of the waves by the
terminal resistance will be complete. We may regard this
as the critical resistance for the circuit, and we shall express
B' in terms of it by putting B'^^Lv. We then have
Y 2 _ _ p + n — mvx
Y l p + n + mvx
Damped WaveTrain. — To pass to the case of a damped
train transmitted from the origin in the positive direction of z
we put
in=—j3 + ia } n=—q + ip.
The difference of potential between the wires at any point
after the head of the wavetrain has reached this point is then
represented by an expression of the form
Vo^a'sin {pt—az).
The velocity of propagation is , the frequency £—, the
27777 a
logarithmic decrement — . If the waves suffered no attenu
P
Damped Electrical Oscillations along Parallel Wires. 299
ation in their passage along the leads we should have
j3zgt = when z= P  3 i. e. /3= ^.
2 a, p
In general, it is plain that ( ft ) measures the attenuation.
Inserting the complex variables in equation (1; we have
v z (ftia) 2 =:{pq^ip){cr — q + ip);
... v ^ a * )=( ff q[p + (T)+p(7 . .... (8)
and
2v 2 aft=p{2qpa); (9)
whence
vW + **)= >/{p* + (qp)*\{p a +{q<rr\. . (10)
Velocity of Propagation and Attenuation. — In actual cases
p and cr are small compared with p. If the damping is con
siderable, q may be comparable with p. Accordingly we
expand the righthand side of (10) in ascending powers of p
and a and solve for va and vft. As far as terms of the third
order in p and cr we find •
^f^ + a ^g#+ cd
Hence the velocity of propagation
and the attenuation
= <7" o 1 P +Q 'i y^" 0 ) 8 + (% 2 y)(p+^)(pQ) 2 , i m
If p = <7, there is no distortion, the velocity of propagation
is v, and the attenuation is ^— or ~ for all frequencies : and
7 Ly bv l
the damping has no effect on these quantities.
We have an interesting particular case when p = cr = q.
Then /3 = 0, and the state of affairs is given by
Y =Y e* t sin (pt— ~\
Here the damping and the attenuation are balanced, so that
300 Mr. W. B. Morton on the Propagation of
the wave train in the wires is at any insfcant^u^Zy simple har
monic throughout.
Numerical Values. — To obtain an estimate of the import
ance of the small terms of (13) and (14) I shall take the
numbers given by Dr. Barton in his last paper (loc. cit.).
Judging from his diagram (fig. 2) in that paper, the ampli
tude of the second positive maximum of the wavetrain is
about half that of the first. This would give
e p =2 or q—^~p^ —
We have roughly
p = 2tt x 35 x 10 6 = 22 x 10 7 , q = 2± x 10 6 ,
B = 695xl0 5 , L=19;
.. ^ = 37x10*, and er = 0.
These values give for the velocity of propagation
v{l 00000035} ,
and for the attenuation
^{1+000091},
so that the corrections are quite negligible. We see from
the expressions (13), (14) that the damping q only affects
the value of the small terms introduced by the inequality of
p and a.
Effect of a Terminal Resistance. — To find the effect on the
incident waves of a resistance (without inductance) inserted
between the ends of the wires, we put in the complex values
in the expression (7) for the reflexionfactor. We then get
V 2 _ (p — q + vfix) +i(p — V*oc)
Vi (—p + q + v/3x)—i(p + vax)
=f+ig, say.
Therefore to an incident wave e ipt corresponds a reflected
wave {f+ig)e i P t \ or, taking real parts, with incident cos/?£ we
have reflected
/cos pt—g §\npt = */f 2 +y i cos (pt + 6),
where
tan<9 = ^,;
Damped Electrical Oscillations along Parallel Wires. 301
so that the change of amplitude is accompanied by a change
of phase.
The values come out
, 2 2 _ (pg + vjBxY+ipvax)*
J ~ ty " (p + q + v&v^+ip + vax)* ^7
x*v 2 (cc* + j3*) + 2xv{/3(qp)+*p} + (q P y+p*
g_ = 2xv{Ppa{qp)}
f xy*{a* + l&).(qp)*p* "
(16)
(17)
In order that there should be complete absorption of the
incident waves it is necessary that the two squares in the
numerator of (15) should vanish separately. This requires
va vB
— = —  — = X, say.
p qp ' J
If we substitute va=:p\ and v/3=(q — p)\ in equations (8)
and (9), and eliminate \ by division, we find the condition
reduces to
(p<r){(q P y+p*}=0, .. p=<r.
Therefore complete absorption is only attainable in the
distortionless case. In general we can only reduce the
reflected amplitude to a minimum.
We can write (16) and (17) in the forms
/2 2 _ ax* — 2hx + b
tanfl== W ^~ A2 (19)
From (18) we see that to any value of reflected amplitude
correspond two values of the terminal resistance, say & l9 a? 2 .
We can show that the corresponding phasedifferences 1? 2
are supplementary.
For from (18) we have
b
^1^2=;
11
.: tan *, + tan *i=2 V«&A«[,£^ + ^TTjJ
— _Q 1 + Ay)(a3? 1 ,r 2 & ) _ n
=2V«6A \ a tfb)(a.*ib) ~ U 
Phil. Mag. S. 5. Vol. 47. No. 286. March 1899. Y
302 Lord Kelvin on the Application of Sellmeier's Dynamical
The minimum reflected amplitude is got when
ffl = # 2 = \/ —
V a
The reflexionfactor is then — , and the phasedifference
s/ab + h
is ^. When #=0 we have complete reflexion with unaltered
phase; with # = co, or the circuit open, we have complete
reflexion with reversed phase. The simultaneous alteration
of amplitude and phase difference brings it about that we
appear to pass continuously from amplitude +1 (x = 0) to
amplitude —1 (<^ = oo ) without passing through amplitude zero.
This apparent anomaly was pointed out to me by Dr. Barton.
Putting in the values of a, a, A, and substituting for va and
v/3 approximate values from (11), (12), we And that the
minimum value of the reflexionfactor s/f 2 \g l is
4:{p 2 + q 2 )\
and that the corresponding value of x is
1 _ q{p(r)
neglecting higher terms in pa.
Numerical Values, — Again using Dr. Barton's numbers we
get for the minimum reflexionfactor the value *0004, and
for the corresponding terminal resistance Lv(\ + '00009).
If, therefore, the terminal resistance be adjusted until the
reflected wave is a minimum, we may, without sensible error,
take this resistance to be Lv, and ignore the reflected train
altogether.
Queen's College, Belfast,
13th October, 1898.
XXIII. Application of Sellmeier's Dynamical Theory to the
Dark Lines D l3 D 2 produced by Sodium Vapour, By
Lord Kelvin, G, C. V. 0„ P.R.S.K*
§ 1. T^OR a perfectly definite mechanical representation of
J? Sellmeier's theory, imagine for each molecule of
sodiumvapour a spherical hollow in ether, lined with a thin
rigid spherical shell, of mass equal to the mass of homo
geneous ether which would fill the hollow. This rigid lining
* Communicated by the Author, having been read before the Eoval
Society of Edinburgh on Feb. 6, 1899. J
Theory to Dark Lines D 1? D 2 p reduced by Sodlu m Vapour. 303
of the hollow we shall call the sheath of the molecule, or briefly
the sheath. Within this put two rigid spherical shells, one
inside the other, each movable and each repelled from the
sheath with forces, or distribution of force, such that the
centre of each is attracted towards the centre of the hollow
with a force varying directly as the distance. These suppo
sitions merely put two of Sellmeier's singleatom vibrators
into one sheath.
§ 2. Imagine now a vast number of these diatomic molecules,
equal and similar in every respect, to be distributed homo
geneously through all the ether which we have to consider as
containing sodiumvapour. In the first place, let the density
of the vapour be so small that the distance between nearest
centres is great in comparison with the diameter of each
molecule. And in the first place also, let us consider light
whose wavelength is very large in comparison with the
distance from centre to centre of nearest molecules. Subject
to these conditions we have (Sellmeier's formula)
v,\2 wit 8 mf
©
where m, m, denote the ratios of the sums of the masses of
one and the other of the movable shells of the diatomic
molecules in any large volume of ether, to the mass of un
disturbed ether filling the same volume; /c, /c / the periods of
vibration of one and the other of the two movable shells of
one molecule, on the supposition that the sheath is held
fixed ; v e the velocity of light in pure undisturbed ether ;
v s the velocity of light of period t in the sodiumvapour.
§ 3. For sodiumvapour, according to the measurements
of Rowland and Bell*, published in 1887 and 1888 (probably
the most accurate hitherto made), the periods of light corre
sponding to the exceedingly fine dark lines D 1? D 2 of the
solar spectrum are '589618 and '589022 of a michronf. The
mean of these is so nearly one thousand times their difference
that we may take
* Rowland, Phil. Mag. 1887, first halfyear; Bell, Phil. Mag. 1888,
first halfyear.
t " Michron " is the name which I have given to a special unit of
time such that the velocity of light is one mikrom of space per michron of
time, the mikrom being one millionth of a metre. The best determi
nations of the velocity of light in undisturbed ether give 300,000 kilometres,
or 3 xlO'4 mikrom?, per second. This makes the michron £x 10 14 of a
second.
Y2
304 Lord Kelvin 'on the Application ofSellmeier's Dynamical
Hence if we put
T =i(« + « y )(l+ i ^ 5 ) ..... (3);
and if x be any numeric not exceeding 4 or 5 or 10, we
have
V 2 fc 2 ~ 2x+V T 2 yc y 2_T 2¥l * ' ' ^'
Using this in (1), and denoting by //, the refractive index
from ether to an ideal sodiumvapour with only the two
disturbing atoms m, m n we find
_ 1000 m 1000 m, ,..
2# + l 2#— 1 v J
whence
f 2 . 1000 t 2 . 1000
©'
§ 4. "When the period, and the corresponding value of x
according to (3), is such as to make fi 2 negative, the light
cannot enter the sodiumvapour. When the period is such as
to make /jl 2 real, the proportion (according to Fresnel, and
according to the most probable dynamics,) of normally incident
light which enters the vapour is
* W (7).
§ 5. Judging from the approximate equality in intensity
of the bright lines D l3 D 2 of incandescent sodiumvapour ;
and from the approximately equal strengths of the very fine
dark lines D b D 2 of the solar spectrum ; and from the ap
proximately equal strengths, or equal breadths, of the dark
lines !>!, D 2 observed in the analysis of the light of an incan
descent metal, or of the electric arc, seen through sodium
vapour of insufficient density to give much broadening of
either line ; we see that m and m, cannot be very different,
and we have as yet no experimental knowledge to show that
either is greater than the other. I have therefore assumed
them equal in the calculations and numerical illustrations
described below.
§ 6. At the beginning of the present year I had the great
pleasure to receive from Professor Henri Becquerel, enclosed
with a letter of date Dec. 31, 1898, two photographs of ano
Theory to Dark Lines D l5 D 2 produced by Sodium Vapour. 305
malous dispersion by prisms of sodiumvapour *, by which 1
was astonished and delighted to see not merely a beautiful and
perfect demonstration of the " anomalous dispersion " towards
infinity on each side of the zero of refractivity, but also an
illustration of the characteristic nullity of absorption and
finite breadth of dark lines, originally shown in Sellmeier's
formula t of 1872 and now, after 27 years, first actually
seen. Each photograph showed dark spaces on the high sides
of the D 1? D 2 lines, very narrow on one of the photographs;
on the other much broader, and the one beside the D 2 nne
decidedly broader than the one beside the D l line ; just as it
should be according to Sellmeier's formula, according to
which also the density of the vapour in the prism must have
been greater in the latter case than in the former. Guessing
from the ratio of the breadths of the dark bands to the space
between their D„ D 2 borders, and from a slightly greater
breadth of the one beside D 2 , I judged that m must in this
case have been not very different from *0002 ; and I calculated
accordingly from (6) the accompanying graphical represen
tation showing the value of 1 , represented by y in fig. 1.
Fi g' !• w=0002.
s
t
~\w
D,
1
I
1
1 I
\L
.
9<
6
9<
)7
9<
>8
9<
>9
A
ymp
tote
?=Q
ooz
_y
y
0,A
iym, t
vtott
>
I0(
t\
f
001
1
002
i
003
A
' 0
3 \
n 1
■>
"
* A description of Professor Becquerel's experiments and results will
be found in Comptes Rendus, Dec. 5, 1898, and Jan. 16, 1899.
t Sellmeier, Pogg. Ann. vol. cxlv. (1872) pp. 399, 520 ; vol. cxlvii.
(1872) pp. 387, 525.
306 Lord Kelvin on the Application o/Sellmeiers Dynamical
Fig. 2 represents similarly the value of 1 for m = '001,
or density of vapour five times that in the case represented
Fig.2. ™=00l.
D,
>r f
9
pfi
9
Q7
9
08
1 ioco
A.
t/m/L
ioie,
^=•001
v
rr—
i
001
1
dd?
\
003
1
004
TK
y=<
\Asy
•npto
*£
\
> ''
Fig. 3.
w=0002.
a
998
999 D
2 1000 C
Fig. 4
, 1001
1002
L.
1
>
mf
if
y'
■
i
99
8
9919 Qj 10
00 (
>, 10
01
10
0?
■■ X
Fig. 5.
m=003.
11 1 1111 o s r "T I I 1
996 997
999 D a 1000 °i 1001 1002
by fig. 1. Figs. 3 and 4 represent the ratio of intensities of
transmitted to normally incident light for the densities corre
sponding to figs. 1 and 2; and fig. 5 represents the ratio for
Theory to Dark Lines D„ D 2 produced by Sodium Vapour. 30?
the density corresponding to the value m=*003. The fol
lowing table gives the breadths of the dark bands for densities
of vapour corresponding to values of m from *0002 to fifteen
times that value ; and fig. 6 represents graphically the breadths
of the dark bands and their positions relatively to the bright
lines D 1? D 2 for the first five values of m in the table.
Values of m.
Breadths of Bands.
D r
D 2 .
•0002
•09
•217
•293
•340
•371
•392
•408
•419
•11
•383
•707
1060
1429
1808
2192
2581
•0006
•0010
•0014
•0018
•0022
■0026
•0030.........
Fig. 6.
VmV£S orM
^n
J D °
002
j^
■ ,(
006
5° 2
_j d '
Pd 2
Jo,
310
{j^~
1 0
014
_J D '
m^
018
§ 7. According to Sellmeier's formula the light transmitted
through a layer of sodium vapour (or any transparent sub
stance to which the formula is applicable) is the same whatever
be the thickness of the layer (provided of course that the
thickness is at least several wavelengths, and that the ordinary
theory of the transmission of light through thin plates is
taken into account when necessary). Thus the D 1? D 2 lines
of the spectrum of solar light, which has travelled from the
source through a hundred kilometres of sodiumvapour in the
sun's atmosphere, must be identical in breadth and penumbras
with those seen in a laboratory experiment in the spectrum of
BOS Lord Rayleigh on the Cooling of Air by Radiation
light transmitted through half a centimetre or a few centi
metres of sodiumvapour, of the same density as the densest
part of the sodiumvapour in the portion of the solar atmo
sphere traversed by the light analysed in any particular
observation. The question of temperature cannot occur except
in so far as the density of the vapour, and the clustering in
groups of atoms, or nonclustering (mist or vapour of sodium),
are concerned.
§ 8. A grand inference from the experimental foundation
of Stokes'' and KirchhofPs original idea is that the periods
of molecular vibration are the same to an exceedingly minute
degree of accuracy through the great differences of range of
vibration presented in the radiant molecules of an electric
spark, electric arc, or flame, and in the molecules of a com
paratively cool vapour or gas giving dark lines in the spectrum
of light transmitted through it.
§ 9. It is much to be desired that laboratory experiments be
made, notwithstanding their extreme difficulty, to determine
the density and pressure of sodiumvapour through a wide
range of temperature, and the relation between density,
pressure, and temperature of gaseous sodium.
XXIV. On the Cooling of Air by Radiation and Conduction,
and on the Propagation of Sound. By Lord Rayleigh,
F.R.S*
ACCORDING to Laplace's theory of the propagation of
Sound the expansions (and contractions) of the air
are supposed to take place without transfer of heat. Many
years ago Sir G. Stokes f discussed the question of the
influence of radiation from the heated air upon the propagation
of sound. He showed that such small radiating power as is
admissible would tell rather upon the intensity than upon the
velocity. If x be measured in the direction of propagation,
the factor expressing the diminution of amplitude is e~ mx ,
where
m=2^1jL (1)
In (1) 7 represents the ratio of specific heats (1*41), a is the
velocity of sound, and q is such that e~^ represents the law of
cooling by radiation of a small mass of air maintained at
constant volume. If t denote the time required to traverse
the distance x } r=xja, and (1) may be taken to assert that
the amplitude falls to any fraction, e. g. onehalf, of its original
* Communicated by the Author.
t Phil. Mag. [4] i. p. 30o, 1851 : Theory of Sound, § 247.
m
and Conduction, ami on the Propagation of Sound. 309
value in 7 times the interval of time required by a mass of
air to cool to the same fraction of its original excess of
temperature. " There appear to be no data by which the
latter interval can be fixed with any approach to precision ;
but if we take it at one minute, the conclusion is that sound
would be propagated for (seven) minutes, or travel over about
(80) miles, without very serious loss from this cause 9> *. We
shall presently return to the consideration of the probable
value of q.
Besides radiation there is also to be considered the influence
of conductivity in causing transfer of heat, and further there
are the effects of viscosity. The problems thus suggested
have been solved by Stokes and Kirchhofff. If the law of
propagation be
u==e m'x cos (nt—x/a), . . . . . (2)
then
'=£{t^}' ^
in which the frequency of vibration is n/"27r, juJ is the kine
matic viscosity, and v the thermometric conductivity. In
c.G.s. measure we may take /// = *14, v=\26, so that
To take a particular case, let the frequency be 256 ; then
since a = 33200, we find for the time of propagation during
which the amplitude diminishes in the ratio of e : 1 ,
(m'a) =3560 seconds.
Accordingly it is only very high sounds whose propaga
tion can be appreciably influenced by viscosity and conduc
tivity.
If we combine the effects of radiation with those of viscosity
and conduction, we have as the factor of attenuation
i
g— (m+m^x
where m + m , = '14:(q/a) + '12 (n?/a?) (4)
In actual observations of sound we must expect the
intensity to fall off in accordance with the law of inverse
squares of distances. A very little experience of moderately
distant sounds shows that in fact the intensity is in a high
degree uncertain. These discrepancies are attributable to
* Proc. Roy. Inst., April 9, 1879.
t Pogg. Ann. vol. cxxxiv. p. 177, 1868 ; Theory of Sound, 2nd ed., § 348.
310 Lord Rayleigh on the Cooling of Air by Radiation
atmospheric refraction and reflexion, and they are sometimes
very surprising. But the question remains whether in a
uniform condition of the atmosphere the attenuation is sensibly
more rapid than can be accounted for by the law of inverse
squares. Some interesting experiments towards the elucida
tion of this matter have been published by Mr. Wilmer Duff *,
who compared the distances of audibility of sounds proceeding
respectively from two and from eight similar whistles. On an
average the eight whistles were audible only about onefourth
further than a pair of whistles ; whereas, if the sphericity of
the waves had been the only cause of attenuation, the dis
tances would have been as 2 to 1. Mr. Duff considers that in
the circumstances of his experiments there was little oppor
tunity for atmospheric irregularities, and he attributes the
greater part of the falling off to radiation. Calculating from
(1) he deduces a radiating power such that a mass of air at
any given excess of temperature above its surroundings will
(if its volume remain constant) fall by radiation to onehalf
of that excess in about onetwelfth of a second.
In this paper I propose to discuss further the question of the
radiating power of air, and I shall contend that on various
grounds it is necessary to restrict it to a value hundreds
of twines smaller than that above mentioned. On this view
Mr. Duff's results remain unexplained. For myself I should
still be disposed to attribute them to atmospheric refraction.
If further experiment should establish a rate of attenuation of
the order in question as applicable in uniform air, it will I
think be necessary to look for a cause not hitherto taken into
account. We might imagine a delay in the equalization of
the different sorts of energy in a gas undergoing compression,
not wholly insensible in comparison with the time of vibra
tion of the sound. If in the dynamical theory we assimilate
the molecules of a gas to hard smooth bodies which are nearly
but not absolutely spherical, and trace the effect of a rapid
compression, we see that at the first moment the increment
of energy is wholly translational and thus produces a maxi
mum effect in opposing the. compression. A little later a due
proportion of the excess of energy will have passed into,
rotational forms which do not influence the pressure, and this
will accordingly fall off. Any effect of the kind must give
rise to dissipation, and the amount of it will increase with the
time required for the transformations, i. e. in the above men
tioned illustration with the degree of approximation to the
spherical form. In the case of absolute spheres no transforma
tion of translatory into rotatory energy, or vice versa, would
* Phys. Keview, vol. vi. p. 129,.189B.
and Conduction, and on the Propagation of Sound. 311
occur in a finite time. There appears to be nothing in the
behaviour of gases, as revealed to us by experiment, which
forbids the supposition of a delay capable of influencing the
propagation of sound.
Returning now to the question of the radiating power of air,
we may establish a sort of superior limit by an argument based
upon the theory of exchanges, itself firmly established by the
researches of B. Stewart. Consider a spherical mass of radius
r, slightly and uniformly heated. Whatever may be the
radiation proceeding from a unit of surface, it must be less
than the radiation from an ideal black surface under the same
conditions. Let us, however, suppose that the radiation is the
same in both cases and inquire what would then be the rate
of cooling. According to Bottomley* the emissivity of a
blackened surface moderately heated is '0001. This is the
amount of heat reckoned in watergramdegree units emitted
in one second from a square centimetre of surface heated 1° C.
If the excess of temperature be 0, the whole emission is
0x47n' 2 xOOOl.
On the other hand, the capacity for heat is
7r>> 3 x0013x24,
the first factor being the volume, the second the density^ and
the third the specific heat of air referred as usual to water.
Thus for the rate of cooling,
dd 0003 1
Jdt = ""•0013x«24xr = ~ r ^ nea * l 7>
whence 0=0 Qe V r , (5)
O being the initial value of 6. The time in seconds of cooling
in the ratio of e : 1 is thus represented numerically by r
expressed in centims.
When r is very great, the suppositions on which (5) is
calculated will be approximately correct, and that equation
will then represent the actual law of cooling of the sphere of
air, supposed to be maintained uniform by mixing if neces
sary. But ordinary experience, and more especially the
observations of Tyndall upon the diathermancy of air, would
lead us to suppose that this condition of things would not be
approached until r reached 1000 or perhaps 10,000 centims.
For values of r comparable with the half wavelength of
ordinary sounds, e.g. 30 centim., it would seem that the real
time of cooling must be a large multiple of that given by (5) .
* Everett, C.G.S, Units, 1891, p. 134.
312 Lord Rayleigh on the Cooling of Air by Radiation
At this rate the time of cooling of a mass of air must exceed,
and probably largely exceed, 60 seconds. To suppose that
this time is onetwelfth of a second would require a sphere of
air 2 millim. in diameter to radiate as much heat as if it were
of blackened copper at the same temperature.
Although, if the above argument is correct, there seems
little likelihood of the cooling of moderate masses of air being
sensibly influenced by radiation, I thought it would be of
interest to inquire whether the observed cooling (or heating)
in an experiment on the lines of Clement and Desormes
could be adequately explained by the conduction of heat from
the walls of the vessel in accordance with the known con
ductivity of air. A nearly spherical vessel of glass of about
35 centim. diameter, well encased, was fitted, airtight, with two
tubes. One of these led to a manometer charged with water or
sulphuric acid ; the other was provided with a stopcock and
connected with an airpump. In making an experiment the
stopcock was closed and a vacuum established in a limited
volume upon the further side. A rapid opening and re
closing of the cock allowed a certain quantity of air to
escape suddenly, and thus gave rise to a nearly uniform
cooling of that remaining behind in the vessel. At the same
moment the liquid rose in the manometer, and the obser
vation consisted in noting the times (given by a metronome
beating seconds) at which the liquid in its descent passed
the divisions of a scale, as the air recovered the temperature
of the containing vessel. The first record would usually be
at the third or fourth second from the turning of the cock,
and the last after perhaps 120 seconds. In this way data
are obtained for a plot of the curve of pressure; and the
part actually observed has to be supplemented by extra
polation, so as to go back to the zero of time (the moment of
turning the tap) and to allow for the drop which might occur
subsequent to the last observation. An estimate, which
cannot be much in error, is thus obtained of the w r hole rise in
pressure during the recovery of temperature, and for the
time, reckoned from the commencement, at which the rise
is equal to onehalf of the total.
In some of the earlier experiments the w r hole rise of
pressure (fall in the manometer) during the recovery of
temperature was about 20 millim. of water, and the time of
half recovery was 15 seconds. I was desirous of working
with the minimum range, since only in this way could it be
hoped to eliminate the effect of gravity, whereby the interior
and still cool parts of the included air would be made to
fall and so come into closer proximity to the walls, and thus
and Conduction, and on the Propagation of Sound. 313
accelerate the mean cooling. In order to diminish the dis
turbance due to capillarity, the bore of the manometertube,
which stood in a large open cistern, was increased to about
18 millim.*, and suitable optical arrangements were intro
duced to render small movements easily visible. By degrees
the raDge was diminished, with a prolongation of the. time of
half recovery to 18, 22, 24, and finally to about 26 seconds.
The minimum range attained was represented by 3 or 4
millim. of water, and at this stage there did not appear to be
much further prolongation of cooling in progress. There
seemed to be no appreciable difference whether the air was
artificially dried or not, but in no case was the moisture
sufficient to develop fog under the very small expansions
employed. The result of the experiments maybe taken to be
that when the influence of gravity was, as far as practicable,
eliminated, the time of half recovery of temperature was
about 26 seconds.
It may perhaps be well to give an example of an actual
experiment. Thus in one trial on Nov. 1, the recorded times
of passage across the divisions of the scale were 3, 6, 11, 18,
26, 35, 47, 67, 114 seconds. The divisions themselves were
millimetres, but the actual movements of the meniscus were
less in the proportion of about 2J : 1. A plot of these
numbers shows that one division must be added to represent
the movement between s and 3 s , and about as much
for the movement to be expected between 114 s and oo .
The whole range is thus 10 divisions (corresponding to
4 millim. at the meniscus), and the mid point occurs at
26 s . On each occasion 3 or 4 sets of readings were taken
under given conditions with fairly accordant results.
It now remains to compare with the time of heating
derived from theory. The calculation is complicated by
the consideration that wdien during the process any part
becomes heated, it expands and compresses all the other
parts, thereby developing heat in them. From the investi
gation which follows f, we see that the time of half recovery t
is given by the formula
' — ^r> < 6 )
in which a is the radius of the sphere, y the ratio of specific
heats (1*41), and v is the thermometric conductivity, found
by dividing the ordinary or calorimetric conductivity by the
* It must not be forgotten that too large a diameter is objectionable,
as leading to an augmentation of volume during an experiment, as the
liquid falls.
t See next paper,
314 Lord "Rayleigh on Conduction of Heat in a Spherical
thermal capacity of unit, volume. This thermal capacity is
to be taken with volume constant, and it will be less than the
thermal capacity with pressure constant in the ratio of 7 : 1.
Accordingly v/y in (6) represents the latter thermal capacity,
of which the experimental value is "00128 x *239, the first
factor representing the density of air referred to water.
Thus, if we take the calorimetric conductivity at '000056. we
have in c.G.s. measure
j>='258, v/ 7 =183;
and thence
^ = 102« 2 .
In the present apparatus a, determined by the contents, is
16*4 centim., whence
£=27*4 seconds.
The agreement of the observed and calculated values is
quite as close as could have been expected, and confirms the
view that the transfer of heat is due to conduction, and that
the part played by radiation is insensible. From a com
parison of the experimental and calculated curves, however,
it seems probable that the effect of gravity was not wholly
eliminated, and that the later stages of the phenomenon, at
any rate, may still have been a little influenced by a downward
movement of the central parts.
XXY. On tlie Conduction of Heat in a Spherical Mass of Air
confined by Walls at a Constant Temperature. By Lord
Rayleigh, F.R.S.*
IT is proposed to investigate the subsidence to thermal
equilibrium of a gas slightly disturbed therefrom and
included in a solid vessel whose walls retain a constant
temperature. The problem differs from those considered by
Fourier in consequence of the mobility of the gas, which
may give rise to two kinds of complication. In the first
place gravity, taking advantage of the different densities
prevailing in various parts, tends to produce circulation. In
many cases the subsidence to equilibrium must be greatly
modified thereby. But this effect diminishes with the amount
of the temperature disturbance, and for infinitesimal dis
turbances the influence of gravity disappears. On the other
hand, the second complication remains, even though we limit
ourselves to infinitesimal disturbances. When one part of
the gas expands in consequence of reception of heat by
* Communicated by the Author.
Mass of Air confined by Walls at Constant Temperature, 315
radiation or conduction, it compresses the remaining parts,
and these in their turn become heated in accordance with the
laws of gases. To take account of this effect a special
investigation is necessary.
But although the fixity of the boundary does not suffice to
prevent local expansions and contractions and consequent
motions of the gas, we may nevertheless neglect the inertia of
these motions since they are very slow in comparison with
the free oscillations of the mass regarded as a resonator.
Accordingly the pressure, although variable with time, may
be treated as uniform at any one moment throughout the mass.
In the usual notation *, if .9 be the condensation and 6 the
excess of temperature, the pressure p is given by
p = kp(l + s + aO). . (1)
The effect of a small sudden condensation s is to produce an
elevation of temperature, which may be denoted by /3s. Let
rfQ be the quantity of heat entering the element of volume in
the time dt, measured by the rise of temperature which it
would produce, if there were no " condensation/' Then
d0  R ds j. d $ i'9\
JtPd; + it* {Z)
and, if the passage of dQ be the result of radiation and con
duction, we have
§**&. .... • (3)
In (3) v represents the " thermometric conductivity h found
by dividing the conductivity by the thermal capacity of the
gas (per unit volume), at constant volume. Its value for air
at 0° and atmospheric pressure may be taken to be *26
cm 2 ./sec. Also q represents the radiation, supposed to depend
only upon the excess of temperature of the gas over that of
the enclosure.
If</Q=0, 0^/38, and in (1)
p=A/){l + (l + ^};
so that
i + «£=y, (4)
where y is the wellknown ratio of specific heats, whose value
for air and several other gases is very nearly 1*41.
In general from (2) and (3)
i+4***'* • • • • •?>
*
'Theory of Sound,' §247.
316 Lord Rayleigh on Conduction of Heat in a Spherical
In order to find the normal modes into which the most
general subsidence may be analysed, we are to assume that s
and 6 are functions of the time solely through the factor e~ w .
Since p is uniform, s\a0 must by (1) be of the form H e~ h \
where H is some constant ; so that if for brevity the factor
e~ M be dropped,
s + «<9=H; ....... (6)
while from (5)
vS7 2 0+{hq)0 = hi3s (7)
Eliminating s between (5) and (7), we get
V s + m 2 (0C)=.O, ..... (8)
where
hy — q n 7</3H
, ' 0J^. • • • • 0)
These equations are applicable in the general case, but when
radiation and conduction are both operative the equation by
which m is determined becomes rather complicated. If there
be no conduction, v = 0. The solution is then very simple,
and may be worth a moment's attention.
Equations (6) and (7) give
0= kpR , = (H)H, . . . (10)
hy — q hy — q v '
Now the mean value of s throughout the mass, which does
not change with the time, must be zero ; so that from (10) we
obtain the alternatives
(i.) h = q, (ii.) H = 0.
Corresponding to (i.) we have with restoration of the time
factor
0=(H/a>««, s=0. .... (11)
In this solution the temperature is uniform and the condensa
tion zero throughout the mass. By means of it any initial
mean temperature may be provided for, so that in the
remaining solutions the mean temperature may be considered
to be zero.
In the second alternative H = 0, so that s=—a6. Using
this in (7) with v evanescent, we get
(hyq)d=0 (12)
The second solution is accordingly
0=<j>{x,y,z)e«% s=x<l>fay,z)eMr, . (13)
where <£ denotes a function arbitrary throughout the mass,
except for the restriction that its mean value must be zero.
Mass of Air confined by Walls at Constant Temperature. 317
Thus if <£) denote the initial value of 6 as a function of
x, y, z, and O its mean value, the complete solution may be
written
0=0 o «8'+(00 o )<r«'/v ]
k . . . (14)
s = « a (@_0 o ) £? ^/y J
giving
* + «0=a@ o <r8< (15)
It is on (15) that the variable part of the pressure depends.
When the conductivity v is finite, the solutions are less
simple and involve the form of the vessel in which the gas is
contained. As a first example we may take the case of gas
bounded by two parallel planes perpendicular to x, the
temperature and condensation being even functions of x
measured from the midplane. In this case \J" 2 = a n ldx 2 , and
we get
= Cf Acosma?, s/« = D + Acosw^, . . (16)
S + a0 = «CaD = H (17)
By (9), (17)
CM!, D=i!LpM v . . . (18^
hy — q a [liy — q)
There remain two conditions to be satisfied. The first is
simply that = when x = + «, 2a being the distance between
the walls. This gives
C + Acos?na = (19)
The remaining condition is given by the consideration that
the mean value of s, proportional to \sdx, must vanish.
Accordingly
ma.D+ sin ma. A = (20)
From (18), (19), (20) we have as the equation for the
admissible values of m,
tan ma ctfiq — vm 2
ma a(3{q + vm i y
reducing for the case of evanescent q to
tan ma 1
ma aj3
The general solution may be expressed in the series
Phil. Mag. S. 5. Yol. 47. No 286. March 1899.
(21)
(22)
(23)
318 Lord Rayleigh on Conduction of Heat in a Spherical
where h ly h^ . . . are the values of 7i corresponding according
to (9) with the various values of m, and ft, ft . . . are of the
form
1 = cosm l x— cos m x a ~\
>>•••• (24)
s 1 = — a(cos m±x — sin m^a/w^a) J
It only remains to determine the arbitrary constants Af,
A 2 , . t . to suit prescribed initial conditions. We will limit
ourselves to the simpler case of q = 0, so that the values
of m are given by (22). With use of this relation and
putting for brevity a=l, we find from (24)
ftft dx= 3— cos nil cos m 2 ,
a P
1 j a ^ + l
5 X 5 2 ax = 32 — cos m x cos m 2 ;
'0 P
so that
( 1 2 dx + /3/a.\ sfrdv^O, .... (25)
ft, ft being any (different) functions of the form (24). Also
'^^^^{U.^}. . (26)
There is now no difficulty in finding A 1? A 2 , . . . to suit
arbitrary initial values of and its associated s, i. e. so that
e=A 1 ft + A,0 2 + ..
(27)
t
Jo
f
Jo
S
Thus to determine A 1?
=A 1 ft + A a ft+ ... I
= A 1 5 1 +A 2 5 2 + ...
f 1 (® ft + pl'a . Ssj)^ = A 1 C (ft 2 + £/« . Sl 2 )dx
+ A 2 (0 1 2 +l3/*.s 1 s 2 )da!+..
Jo
in which the coefficients of A 2 , A 3 . . . vanish by (25) ; so that
by (26)
M i+ ^1 = i^Jo 1(0<?i+/3a * s ^ • (28)
An important particular case is that in which © is constant,
and accordingly S = 0. Since
f 1 sin 7^ l + a/3
1 V 1 dX= C0SWi = — ■ tt^COSWi,
Jo ™i «P
(29)
Mass of Air confined by Walls at Constant Temperature. 3J9
we have
a __ 2® cos m x
a/3 + cos 2 /^
For the pressure we have
0+s/* = A 1 eK(cosm l +^^) + ..".".
= 7j— cos m, . A x e V +
or in the particular case of (29),
d + sa = 2® — 3^ 1 . + (30)
afj a/3 + COS 2 mi v '
If /3=0, we fall back upon a problem of the Fourier type.
By (22) in that case
ma = ^7r(l, 3, 5, . . .)
cos 2 ma = a?fi 2 /m 2 a 2 ,
so that (30) becomes
20 fe + ^v + > • • • ^
or initially
80/1 X , 1 . \ . _
^(p+ 35+52 + ...) »,«.e.
The values of /* are given by
/i= ^ (12 ' 32 ' 52 ') ( 32 )
We will now pass on to the more important practical case
of a spherical envelope of radius a. The equation (8) for
(0—C) is identical with that which determines the vibrations
of air * in a spherical case, and the solution may be expanded
in Laplace's series. The typical term is
d(3={mr)hJ nH (mr).Y n , . . . (33)
Y n being the surface spherical harmonic of order n where
n=0, 1, 2, 3 . . ., and J the symbol of Bessel's functions. In
virtue of (6) we may as before equate — s/a — D, where D is
another constant, to the righthand member of (33) . The two
conditions yet to be satisfied are that 0=0 when >*=a, and
that the mean value of s throughout the sphere shall vanish,
* l Theory of Sound/ vol. II. cli. xvii.
Z2
320 Lord Rayleigh on Conduction of Heat in a Spherical .
When the value of n is greater than zero, the first of these
conditions gives C = and the second D = 0; so that
6=s/cc=(mr)»J nH (mr) .Y n , . . . (34)
and s + <x0 = O. Accordingly these terms contribute nothing
to the pressure. It is further required that
J n+i (ma)=0, (35)
by which the admissible values of m are determined. The
roots of (35) are discussed in ' Theory of Sound,' § 206 . . . ;
but it is not necessary to go further into the matter here, as
interest centres rather upon the case n = 0.
. If we assume symmetry with respect to the centre of the
1 d*
sphere, we may replace V 2 in (8) by  y^ r, thus obtaining
^fr«" 0) +™MflC)=0, . . . (36)
of which the general solution is
n fi . cos mr ^ sin mr
u = \u f A f r>
mr mr
But for the present purpose the term r~ l cos mr is excluded,
so that we may write
= Q + B™2*,  S /«=D + B™', . (37)
mr mr
giving
5 + a0 = a(CD) = H. . . . {37 bis)
The first special condition gives
7waC + Bsinma = (38)
The second, that the mean value of s shall vanish, gives on
integration
im 3 tt 3 D 4 B(sin ma — ma cos ma) = 0. . . (39)
Equations (18), derived from (9) and (37 his), giving C
and D in terms of H, hold good as before. Thus
D _ gh _ a/Bqvm 9 
C ~ hafi ~ a/3{qivm 2 ) { }
Equating this ratio to that derived from (38), (39), we find
3 ma cos ma — sin ma _ vni 2 — aftq
m?a*  sin ma a/3 (vm 2 + q) ' ^ '
Mass of Air confined by Walls at Constant Temperature. 321
This is the equation from which m is to be found, after which
h is given by (9).
In the further discussion we will limit ourselves to the case
°f 9 = ®> when (41) reduces to
m
3a{3{mcotml), .... (42)
in which a has been put equal to unity. Here by (40)
D=C/«/3.
Thus we may set as in (23),
*B^ + B,r** + ) ■ m >
s = B l e^ t s 1 +B 3 « V^ 2 + )
in which
Q $\\\m x r sinra^ sin m,r 1 sin mi a ,...
Vi= , Si=— a — —,(44)
m x r m Y a m x r /3 m x a v '
and by (9) h l = vm 2 l /y. Also
, , n 1 + a/3 sin m\a ,._,.
S a + 1 = ^ l  (45)
The process for determining B b B 2 , . . . . follows the same
lines as before. By direct integration from (44) we find
2W!W 2 ( l
2 + /3/a . s^s 2 )r % dr
_ sin (m, — m 2 ) sin (m 1 f m 2 ) 2 sin m 1 sinw? 2
mi—m 2 W]+m 2 3a/3
a being put equal to unity. By means of equation (42)
satisfied by m 1 and m 2 we may show that the quantity on the
right in the above equation vanishes. For the sum of the
first two fractions is
2??7 2 sin m x cos in 2 — 2m 3 sin m 2 cos m x
?><, — w*
of which the denominator by (42) is equal to
Sa^{m l cot m 1 — m 2 cot m 2 ).
ingly \\e&
Jo
Accordingly {0& + P/* . s&ydr^Q. . . . (46)
322 Lord Rayleigh on Conduction of Heat in a Spherical
Also
2V C l , a9 , ot » *j i sin 2wii , 2 sin 2 m x ( „.
To determine the arbitrary constants B 1 . . . . from the given
initial values of 6 and 5, say © and S, we proceed as usual.
AVe limit ourselves to the term of zero order in spherical
harmonics, i. e. to the supposition that 6, s are functions of r
only. The terms of higher order in spherical harmonics, if
present, are treated more easily, exactly as in the ordinary
theory of the conduction of heat. By (43)
6 = Bi#i + B 2 2 + i
S=B l5l +B 2 5 2 + ...
and thus
P (e^ + ZS/a . $ Sl )r*dr= B x ( * (0* + 0/a . s^Alr
Jo Jo
+ B 2 \ l [6 A + PI a . S&) t*dr + ....,
Jo
in which the coefficients of B 2 , B 3 , vanish by (46). The
coefficient of Bx is given by (47) . Thus
9rn 2 C 1
I sin 2m a 2 sin 2 m l
^ l \ 1 ~¥inT+~~^afi~
(49)
by which B} is determined. v '
An important particular case is that where © is constant
and accordingly 8 vanishes. Now with use of (42)
C l fl <2rf , _ Sm 1Ul ~~ ??l ! C0S 7Ul Sm 7??1 _ (1 "*" a fi) Sm ??l !
J l . " mj 3 3w l 3a^m l
so that
t, f . sin 2 m! , 2 sin 2 wit] 2j»i sin m. . @ ._.
B '( 1 ^r + 3^} : 3^ — •• ^
Bj, B 2 , .... being thus known, 6 and 5 are given as functions
of the time and of the space coordinates by (43), (44).
To determine the pressure in this case we have from (45)
+ s/a _l + afiy^ sm 2 m.e ht ,.,.
©
a$ jU &*£/ sin2m\' '
Mass of Air confined by Walls at Constant Temperature. 323
the summation extending to all the values of m in (42).
Since (for each term) the mean value of s is zero, the right
hand member of (51) represents also #/©, where 6 is the
mean value of 0.
If in (51) we suppose /3=0, we fall back upon a known
Fourier solution, relative to the mean temperature of a
spherical solid which having been initially at uniform tempe
rature ® throughout is afterwards maintained at zero all over
the surface. From (42) we see that in this case sin m is small
and of order ft. Approximately
sin m=3aft/m ;
and (51) reduces to
of which by a known formula the righthand member iden
tifies itself with unity when £ = 0. By (9) with restoration
of a,
/ i= (l 2 , 3 2 , 5 2 , ....>7r 2 /a 2 (53)
In the general case we may obtain from (42) an approxi
mate value applicable when m is moderately large. The
first approximation is m = iir, i denoting an integer. Suc
cessive operations give
. t 3a ft 18* 2 /3 2 + 9* 3 /3 3 ,..
™=^+ 1  ^ • • (54)
In like manner we find approximately in (51)
sin 2 m (l+aft)/*ft _ 6(l + *g) f , _ 15*/3 + 9* 2 / 3 2 V
i 2 7r 2 y
.... (55)
showing that the coefficients of the terms of high order in (51)
differ from the corresponding terms in (52) only by the factor
(1 + aft) or 7.
In the numerical computation we take 7 = 1'41 , a/3 = '41.
The series (54) suffices for finding m when i is greater than 2.
The first two terms are found by trial and error with trigo
nometrical tables from (42) . In like manner the approximate
value of the lefthand member of (51) therein given suffices
when i is greater than 3. The results as far as / = 12 are
recorded in the annexed table.
. , 'daftf sin 2mA
324 On the Conduction of 'Heat in a Spherical Mass of Air.
■
mJTT.
Lefthand
member
of (55).
•4942
•1799
•0871
•0510
•0332
•0233
!
i.
mjf'ir.
Lefthand
member
of (55).
1
10994
20581
30401
40305
50246
60206
•3
10
n
12
70177
80156
90138
100125
110113
12 0104
0175
•0134
•0106
•0086
•0071
•0060
2
3
4
5
6
Thus the solution (51) of our problem is represented by
^/e = 4942^ 1 °" 4 ^' + 1799^ 20581 ^ , + . .. . . (56)
where by (9) , with omission of g and restoration of a,
t f lt = iT 2 v^a 2 (57)
The numbers entered in the third column of the above
table would add up to unity if continued far enough. The
verification is best made by a comparison with the simpler
series (52). If with t zero we call this series 2' and the
present series 2, both 2 and 2' have unity for their sum, and
accordingly yL' — 2 = 7 — 1, or
111
7
£(j+£+i.+v).*"
1=41.
Here 67/71 2 = '8573, and the difference between this and the
first term of 2, i. e. *4942, is "3631. The differences of the
second, third, &c. terms are '0344, '0082, '0026, 0011, '0005,
♦0000, &c, making a total of '4099.
P.
(fO).
f.
,56).
•00
1 0000
•7037
•6037
•4811
•4002
•3401
•2926
•60 1
•70
2538
2215
1940
1705
1502
0809
0441
•05
•10
•80
•20
•30
•90
100 !
150
200
•40
•50
i 1
We are now in a position to compute the righthand
Notices respecting New Books. 325
member of (56) as a function of t f . The annexed table con
tains sufficient to give an idea of the course of the function.
It is plotted in the figure. The second entry (t / = '05) requires
the inclusion of 9 terms of the series. After t'='7 two terms
suffice; and after t'='2'0 the first term represents the series
to four places of decimals.
By interpolation we find that the series attains the value *5
when
^=•184. ...... (58)
XXVI. Notices respecting New Books.
An Elementary Course in the Integral Calculus. By Dr. D. A.
Murray, Cornell University. Longmans, 1898. Pp. x + 288.
r^E. MUEEAY states his object to be to present " the subjeet
*^ matter, which is of an elementary character, in a simple
manner." This he has succeeded in doing, and the work is well
arranged and the explanations given are exceedingly clear. In
Chapter I. he treats Integration as a process of summation, and in
Chapter II. as the inverse of differentiation. The author's object
herein is to give the student a clear idea of what the Integral Calculi s
is, and of the uses to which it may be applied. The first ten chapters
are devoted to a treatment of the matters handled in such works as
Williamson's, Edwards's, and other wellknown treatises. Chap
ter XL treats of approximate integration, and the application of
the Calculus to the measurement of areas. Here we have clear
statements and proofs of the trapezoidal rule, Simpson's one
third rule, aud Duraud's rule. To this latter gentleman the
author is indebted for valuable suggestions of use to engineering
students. Prof. Durand has also put at Dr. Murray's disposal
his article on " Integral Curves " (in the • Sibley Journal of
826 Geological Society : —
Engineering/ vol. xi. no. 4*), and his account of the ' Fundamental
Theory of the Planimeter 'f . Chapter XIII. is devoted to ordinary
differential equations. The Appendix (some 50 pages) discusses
some of the matters in the text at greater length than is required
by the elementary student, and also contains a large collection of
the figures of the curves which are referred to in the exercises.
Answers are given to these exercises, and further there is a full
Index, and numerous brief historical notes which add to the
utility of an excellent textbook. Dr. Murray suitably acknow
ledges his great indebtedness to his predecessors in the same field.
The book is very neatly and correctly printed, and is of handy size.
XXVII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from vol. xlvi. p. 508.]
November 9th, 1898.— W. Whitaker, B.A., F.E.S., President,
in the Chair.
rpHE following communications were read : —
* 1. " On the Palaeozoic Eadiolarian Eocks of New South Wales.'
By Prof. T. W. Edgeworth David, B.A., F.G.S., and E. F. Pittman,
Esq., Assoc. R.S.M., Government Geologist, New South Wales.
2. ' On the Radiolaria in the Devonian Eocks of New South
Wales.' By G. J. Hinde, Ph.D., F.E.S., F.G.S.
November 23rd.— W. Whitaker, B.A., F.E.S., President,
in the Chair.
The following communications were read : —
1. ' Note on a Conglomerate near Melmerby (Cumberland;.' By
J. E. Marr, Esq., MA., F.E.S., F.G.S.
In this paper the author describes the occurrence of a con
glomeratic deposit which shows indubitable effects of earth
movement, not only on the included pebbles, bat also on the
surface of one of the deposits. The rocks are coloured as basement
Carboniferous rocks on the Geological Survey map. The Skiddaw
Slates are succeeded by about 30 feet of a roughly stratified con
glomerate, followed by 20 to 30 feet of rock with small pebbles, and
that by a second coarse conglomerate. The pebbles possess the
outward form of glacial boulders, but many of them are slicken
sided, fractured, faulted, and indented. The striae are often
curved, parallel, and covered by mineral deposit ; the grains of
the matrix are embedded in the grooves, while slickensiding often
occurs beneath the surface of the pebbles and the striae are seen
to begin or end at a faultplane. The surface of rock beneath
. * Chapter XII. of the work before us is almost a reproduction of this
article, as is also Appendix G which supplements the account in the text,
t Here suitable reference is made to Prof. Henrici's Report (British
Association, 1894) and to Prof. Hele Shaw's paper on 'Mechanical
Integrators ' (' Proceedings of Institution of Civil Engineers, ' vol. lxxxii.
1885).
The Geological Structure of the Southern Mdlverns, Sfc. 327
the upper conglomerate was found to be slickensided. The way in
which the surfaces of some of the pebbles have been squeezedoff
suggests the possibility that their angular shape may be partially
or wholly due to earthmovement.
2. ' Geology of the Great Central Railway (New Extension to
London of the Manchester, Sheffield & Lincolnshire Railway) :
Rugby to Catesby.' By Beeby Thompson, Esq., F.G.S., F.C.S.
In this paper the portion of the line, 10 miles in length, from
Catesby to Rugby is described ; as the ground falls while the strata
rise in this direction, quite low beds in the Lower Lias are met
with near Rugby. The lowest zone exposed is that of Ammonites
semicostatus, in the lower part of which, and in Boulder Clay derived
from it, A. Turneri has been found. The next succeeding zone, that
of A. ohtusus, although for the most part barren, yielded the charac
teristic fossils at its base. The o.vynotuszone is well developed and
well displayed, besides being richly fossiliferous. The zone of
A. raricostatus merges into that of A. oxynotus below and that of
A. armatus above, and is not more than 3 or 4 feet thick. The
annatuszone, beds between that and the Jameso?iizoTie, and the
Jamesonizone itself follow ; the middle beds of the latter being
rich in Rhynchonella and A. pettus, the name of this ammonite is
attached to the zone bearing them. The Ibexzone occurs east of
Flecknoe, covered by rocks yielding A. Eenleyi ; and the highest
beds of this cutting appear to belong to the capricornuszone.
Lists of the characteristic fossils of each zone are given, followed
by a complete list of all those found in the Lower and Middle Lias
of the cuttings, with a statement of their distribution.
The Glacial deposits are described under the following headings : —
Blue or local Boulder Clay, brown and grey contorted Boulder Clay,
Chalky Lower Boulder Clay, (Mid Glacial) sands and gravels, and red
Upper Boulder Clay.
The paper is accompanied by a measure 1 section along the
railway.
3. ' On the Remains of Amia from Oligocene Strata in the Isle
of Wight.' By E. T. Newton, Esq., E.R.S,, F.G.S.
December 7th.— W. Whitaker, B.A., F.R.S., President;
in the Chair.
The following communication was read : —
1. 'The Geological Structure of the Southern Malverns and
of the adjacent District to the West.' By Prof. T. T. Groom,
M.A., D.Sc, F.G.S.
The Raggedstone and Midsummer Hills, consisting ess3ntially of
massive gneissic and schistose rocks, are traversed by a curved
depression which marks a line of profound dislocation, probably of
the nature of a thrustplane. This appears to dip towards the east,
though with a relatively small hade. Along this depression occur
strips of Cambrian and Silurian strata embedded in the Archaean
massif, and indicating the presence of a deep and narrow dis
located synclinal fold. In places, the foliation of the schists
328 Geological Society,
shows a marked relation to the direction of this line, indicating
in all probability a production of schists from the old material in
postLower Palaeozoic times.
The western boundary of the Archaean massif is everywhere a
fault, apparently a thrustplane, but with a small hade. The
direction of this plane is in close relation with the axis of the over
fold into which the Cambrian rocks are thrown to the west of these
hills.
The western boundary of Chase End Hill is likewise a fault,
which is probably a thrustplane with a tolerably low dip towards
the eastern side. The thrust here also appears to have been
accompanied by a secondary production of schists from the old
gneissic series ; and the Cambrian strata are overthrown in the
vicinity of the fault.
There is no evidence for the overlap of the Cambrian Series sup
posed by Holl, the circumstance that the various zones of the
Cambrian Series strike up against the Archaean axis being due to
faulting.
The Cambrian is represented by the following series : —
; Upper Grey Shales.
Coal Hill Igneous Band.
Lower Grey Shales.
Middle Igneous Band.
; Upper Black Shales.
Upper WhiteleavedOak Igneous Band.
Lower Black Shales.
Lower WhiteleavedOak Igneous Baud.
2. Hollybusii Sandstone.
i. hollybush quartzite and conglomerate.
Possils are abundant in certain zones of each of the four sub
divisions of the series.
The Grey Shales rest conformably on the Black Shales, but the
mutual relations of the remaining subdivisions can be decided only
by inference, the junctions being apparently everywhere faults.
The junction between the Cambrian and Archaean is likewise a
fault.
All four divisions of the Cambrian Series are invaded by small
igneous bosses, laccolites, and intercalated sheets of diabase and
andesitic basalt. These igneous rocks do not penetrate the May
Hill Series.
The May Hill Beds seem to rest with apparent conformity upon
the Grey Shales, and do not transgress across the various Cambrian
zones on to the Archaean in the manner hitherto supposed, the
presumed outliers being small patches faulted into the Cambrian.
The structure of the district is to be explained on the supposition
that we are dealing with the western margin of an old mountain
chain overfolded towards the west ; the eastern portion of this
range lies faulted down and buried beneath the Permian and
Mesozoic of the Vale of Gloucester. All the characteristics of a
folded chain are present, namely, the profound folds, overfolds,
thrustplanes, and transverse faults ; and a typical Austonungs
zone is seen to the west.
[ 329 ]
XXVIII. Intelligence and Miscellaneous Articles.
ON THE HEAT PKODUCED BY MOISTENING PULVERIZED BODIES.
NEW THERMOMETR1CAL AND CALORTMETRICAL RESEARCHES.
BY TITO MARTINI *.
TN my second paper presented, last April, at the R. Istitufco
* Yeneto, I dealt with the calorific phenomena observed in
moistening pulverized bodies. The method of experiments has
already been described f ; that is to say, of an arrangement whereby
the liquid ascended to the powder, thoroughly dried, placed in a
glass tube separated from it by a piece of light linen cloth. In
this second series of experiments some modifications were intro
duced in order to keep the powder as dry as possible till the be
ginning of the experiment. The thermometerbulb was placed in
contact with the upper stratum of powder, it having been noticed that
the increase of temperature was more marked from layer to layer.
In the following tables are indicated some of the principal
results obtained with pure charcoal and pure silica (Si0 2 ).
Pure Charcoal (gr. 25).
Name of the Liquid.
Te »p.
of the
Air.
Temp,
of the
Liquid.
Temp.
of the
Charcoal.
Max.
Temp.
Increase
of
Temp.
Liquid
absorbed
in cm. 3
35 cm. 3
30
28
32
34
29
Distilled water
Absolute alcohol
Sulphuric ether
o
9
1916
1071
19 30
2010
710
825
1910
1110
1942
2015
612
c
9 20
1929
1089
1960
2030
720
3430
4505
2948
4460
3360
2890
2510
2576
1859
2500
1330
2170
Acetic ether
Benzene
Bisulphide of carbon.
Name of the Liquid.
Temp,
of the
Air.
Temp,
of the
Liquid.
Temp,
of the
Silica.
Max.
Temp.
Increase
of the
Temp.
Liquid
absorbed .
Distilled water ..
Absolute alcohol
Sulphuric ether
1900
1927
1465
2019
1950
1884
18 96
1887
1460
2025
19 49
1895
1930
I960
1469
2070
19 60
1932
4190
4575
4621
5080
3170
3105
2260
2615
3152
3055
1210
1173
37 cm. 3
36
35
7
40
38
Acetic ether
Benzene
Bisulphide of carbon. . .
1 made also many experiments in order to determine the number
of calories produced by moistening powder. I adopted a special
* " Intorno ad calore che si sviluppa nel bagnare le polveri. Nuove
ricerche termometriche e calorimetriche." Atti del H. Istituto Yeneto,
t. 9, serie vii. p. 927.
t Atti del R. Istituto Veneto, t. 8, serie vii. p. 502; Phil. Mag.
vol. xliv., August 1897.
330
Intelligence and Miscellaneous Articles.
calorimeter made of thin sheet brass, consisting of two cylindrical
tubes, one within the other, having a common axis. The base of
the internal tube was perforated to allow the escape of air, when
pouring the liquid upon the powder with which the tube was
filled. A flannel disk placed at the base prevented the escape of
the powder.
The powder was thoroughly dried before being poured into the
internal tube, which was closed at the top with an indiarubber
stopper, and suspended by three silk threads, within a vessel con
taining chloride of calcium. The external tube contained distilled
water into which was immersed a delicate thermometer.
Calories developed by Charcoal moistened with Distilled Water.
Weight of the
Charcoal.
Volume of the
Water.
Calories, !<*?££■*
44 gr.
40
40
35
30
60 cm. 3
53
58
51
43
62900 1429
56980 1425
57330 1433
51430 1469
44030 1467
Calories developed by Silica moistened with Distilled Water.
Weight of the
Volume of the
Caloriesgr.
Calories by 1 gr.
Silica.
Water.
developed.
of powder.
50 gr.
72 cm. 3
67710
1354
50
70
68450
1369
45
63
60310
1340
40
60
55870
1397
40
62
55870
1397
40
66
55500
1387
35
53
47730
1364
Meissner, in his experiments *, did not use a fixed weight of
water ; at one time he would use a quantity of water equal in
weight to the powder, at another time a quantity double, and at
times much less. In my own case, however, I always used that
quantity of liquid which I found would be absorbed, by capillarity,
by a quantity of charcoal or silica equal to that contained in the
calorimeter. Had I poured on the powder a smaller amount of
water, parts would have remained umnoistened ; a larger quantity
would have absorbed a part of the heat generated.
The foregoing results may be of interest, not only to physicists
in general, but to students of geothermic phenomena. In fact,
the reader will find in my original pamphlet an account of certain
experiments in which silica, moistened with a proportionate
quantity of water, rose from an initial temperature of 19° to that
of 70°.
Venice, June 1898.
* " Ueber diebeiin Benetzen purverfbrniiger Korper auftretecde Warme
tonung." Wiedemann's Annalen, xix. (1886).
Intelligence and Miscellaneous Articles. 331
COMBINATION OF AN EXPERIMENT OF AMPERE WITH AN
EXPERIMENT OF FARADAY. BY J. J. TAUDIN CHABOT.
It was shown by Ampere * how a magnet can be made to rotate
about its axis under the influence of a steady current, and Faraday f
showed how the rotation of a magnet about its axis can give rise
to a steady current +.
By combining these two experiments we obtain a case of
induction by steady currents : a steady primary current in the
circuit E or E x gives rise to a steady secondary current in the
circuit E x or E, a rotating magnet M forming the connecting link.
In order to show the effect, a brake is applied to the magnet M,
a battery is inserted into the circuit E or E 2 and a galvanometer
into the other circuit E 3 or E. On closing the circuit we observe
that the suspended system of the galvanometer remains at zero ;
but, on removing the brake from the magnet, this begins to rotate
and the galvanometer shows a deflection which increases con
tinuously until the magnet turns quite freely. A brake is desirable
which admits of a graduated application.
It appears to me that this experiment is worthy of notice in
consideration of its illustrative character §.
Degerloch (Wiirttemberg),
December 12th, 1898.
EXPERIMENTS WITH THE BRUSH DISCHARGE.
To the Editors of the Philosophical Magazine.
Gentlemen,
I have read with considerable interest the paper by Dr. Cook
on the " Brush Discharge " in the January number of your
Magazine. May I be allowed to call attention to some experiments
made by Lord Blythswood about two years ago, which are of a very
* Recueil d? Observations, p. 177 (1821). Lettre a M. van Beck.
t 'Experimental Researches/ series ii., §§ 217230 (1832); see also
series xxviii. (1851).
X This phenomenon is generally known by the name of " unipolar ''
induction ; " autopolar " induction (induction autopolaire, Gleichpolin
duction), it seems to me, would be better; and therefore 1 propose this
term. Then, in contradistinction, " heteropolar " induction (induction
he"teropolaire, Wechselpolinduction) can be used for signifying induction
by both the poles alternately (dynamo &c).
§ See Phil. Mag. vol, xlvi. p. 428 (Oct, 1808)., and p. o71 (Bee. 1898).
332
Intelligence and Miscellaneous Articles.
similar character. In this case, however, it was the negative glow
which was the source of radiation. The machine used was of the
Wimshurst type, having 160 3ft. plates each carrying 16 sectors.
The experiments were made to determine the effect of the rays
from the negative g]ow on a photographic plate.
Small metallic objects were " radiographed," being placed in
front of, but not touching, a sensitized plate. The whole was then
enclosed in a zinc box (with a small hole cut in the side facing the
source of radiation), which was carefully earthed in order to
prevent any charge on the metallic object affecting the plate. A.
piece of 1 mil. aluminiumfoil was then placed between the source
of radiation and the plate, thus completing the metallic sheath.
The arrangement is shown below.
<Z/nc Box
TO WIMSHURST
MACHINE
PHOTOGRAPH/C
PLATE
/MIL. ALUM'IN/UM
FO/L
"When these precautions were not observed, brush discharges
took place from the points of the metallic object (generally a
small wheel) which strongly affected the plate.
Distinct shadows of the small wheel were obtained after 5
minutes' exposure in a darkened room. It was at first thought that
these photographs were produced by rays similar to the .rrays,
which had traversed the aluminiumfoil ; but it appeared afterwards
that the whole effect was apparently due to minute holes in the
aluminiumfoil, since, when the apparatus was wrapped in black
velvet, no effects were produced on the plate. That such effects were
obtained, however, seems to show that the negative glow possesses
strong actinic power, as shown so conclusively by Dr. Cook.
A possible explanation of the diminished effect observed by
Dr. Cook on an Electroscope placed at a distance from a point at
which a Brush discharge is taking place, when an inductioncoil was
used in place of the Wimshurst machine, seems to be that the
electrification produced would depend on the R.M.S. potential
difference rather than on the maximum value, as indicated by the
sparklength. Similarly for the mechanical force produced by the
wind from the points.
Tours very truly,
Blythswood Laboratory, E. W. Maechajstt.
Renfrew, N.B.
Jan. 30, 1899.
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
\b b a r p;
[FIFTH SERIES.]
APRIL 1899.
XXIX. Longitudinal Vibrations in Solid and Hollow
Cylinders. By C. Chree, Sc.D., LL.D., F.B.S*
Preliminary.
§ 1. TT1HE frequency kfiir of longitudinal vibrations in the
A ideal isotropic bar of infinitely small cross section
has long been known to be given by
k=p K/Efp, (1)
where p is the density, E Young's modulus ; p is given by
<p = i7r/l when the ends are both free or both fixed
p=(2i + l)7r/(2l) when one end is free, the other fixed,
I being the length of the rod and i a positive integer.
For the fundamental or lowest note 2 = 1.
For a circular bar whose radius a, though small compared
to I, is not wholly negligible, the closer approximation
k=p(E/p)i{l}pWa 2 \, (2)
where tj is Poisson's ratio, was obtained independently by Prof.
Pochhammer j" and Lord RayleighJ fully 20 years ago.
§ 2. The subject has been treated by myself in three
papers in the ' Quarterly Journal of . . . Mathematics ' (A)
(p. 287, 1886), (B) (p. 317, 1889), (C) (p. 340, 1890).
In (A) I arrived at (2) describing it (/. c. p. 296) as
* Communicated by the Physical Society : read December 9, 1898.
t Crelle, vol. lxxxi^. (1876).
\ i Theory of Sound,' vol. i. art. 157.
Phil. Mag. S. 5. Vol. 17. No. 287. April 1899. 2 A
334 Dr. C. Chree on Longitudinal
u obtained as a second approximation by Lord Eayleigh." I
further said, " We do not think, however, that his proof affords
any means of judging of the degree of accuracy of the result,
as it is founded on a more or less probable hypothesis and
does not profess to be rigid/' I subsequently learned that
Lord Eayleigh did not admit any want of rigidity in his
proof, and it appears without modification in the second
edition of his Treatise on ' Sound/ I much regret having to
differ from so eminent an authority, but I have not altered
my original opinion.
In (B) I reached the more general result
k=p(V/p)i(li P YK 2 ), (3)
where k is the radius of gyration of the cross section of the
rod about its axis. This was established by a strict elastic
solid method for an elliptic section, and in a somewhat less
rigid way for a rectangular section. (A) and (B) were con
fined, like the investigations of Lord Eayleigh and Professor
Pochhammer, to isotropic materials.
In (C) I considered the more general case of an seolotropic
bar whose long axis was an axis of material symmetry, and
found by strict elastic solid methods that (2) still held for
a circular section, if E denoted Young's modulus for stress
along the length of the bar, and rj Poissou's ratio for the
consequent perpendicular contractions. Further, applying
Lord Eayleigh/ s method, modified in a way I deem necessary,
I obtained (3) for any form of cross section.
§ 3. Since the publication of (( J) Mr. Love has discussed the
subject in vol. ii. of his l Treatise on Elasticity/ On his p. 1 19 *
he refers to (2) as " first given by Prof. Pochhammer . . .
and afterwards apparently independently by Mr. Chree/''
Again, in the new edition of his ' Sound ' Lord Eayleigh,
after deducing (2), says a A more complete solution. . . has
been given by Pochhammer ... A similar investigation has
also been published by Chree.'"'
In view of these remarks, I take this opportunity of stating
explicitly : —
1. That Pochhammer's work was wholly unknown to me
until the appearance of Love's l Elasticity/
2. That my method of solution in (A) is essentially different
from Pochhainmer's, while the methods in (B) and (C) are
absolutely different from his. The method of (A) agrees
with Pochhammer's in employing the equations of elasticity
in cylindrical coordinates. After obtaining, however, — as is
* The preface, p. 13 ; describes the result as " obtained independently "
by me._
Vibrations in Solid and Hollow Cylinders. 335
customary in most elastic problems — the differential equation
for the dilatation, Pochhammer obtains a differential equation
for the quantity
du dw\
1 /du
2\dz~
u and w being the displacements parallel to the radius r and
the axis z, and uses this quantity as a steppingstone to the
values of u and w. On the other hand, I succeeded in
separating u and w so as to obtain at once two differential
equations, in one of which u appeared alone with the dilatation,
while the other contained only w and the dilatation (see (28)
and (29) later).
§ 4. There are two other points in Mr. Love's Treatise to
which I should like to refer. In his art. 263 he substitutes
the term extensional for longitudinal, adding in explanation,
" The vibrations here considered are the ' longitudinal ' vibra
tions of Lord Kayleigh's Theory of Sound. We have
described them as l extensional/ to avoid the suggestion that
there is no lateral motion of the parts of the rod."
I am altogether in sympathy with the object which Mr.
Love has in view (1 expressed myself somewhat strongly on
the point in (A) p. 296, and (C) pp. 3512), but I doubt the
wisdom of attempting to displace a term so generally adopted
as longitudinal.
In the second matter I regret to find myself at variance
with Mr. Love. Referring to transverse vibrations in a rod,
he says on p. 124 of his vol. ii., " the boundary conditions at
free ends cannot be satisfied exactly ... as they can in the . . .
extensional (longitudinal) modes/'' In reality, however, the
boundary equations at a free end are not exactly satisfied in
the case of longitudinal vibrations either by Pochhammer's
solution or my own. The slip may be a purely verbal one on
Mr. Love's part, but his readers might be led to accept the
statement as accurate owing to a slight error in the expression
for the shearing stress zr near the top of Mr. Love's p. 120.
We find there
«r= 2^{ 7 A j J (/c'r) + . . . }e«<7*+?0,
where i= ^ — ±.
The correct expression (compare Mr. Love's second boundary
equation on p. 118) is
^ = ^{2 7 A ^J (ft'r) + . . .}«to # +*>.
2 A2
336 Dr. C. Chree on Longitudinal
Owing to the omission of i in the expression on p. 120 it
looks as if zr vanished for the same values of z as the normal
stress zz. In reality, as I showed in (A) p. 295, zr does not
vanish over a terminal free section of radius a, but is of the
order r(r 2 — a 2 )/l 3 , where r is the perpendicular on the axis.
We are quite justified in neglecting zr when terms of order
(a/1) 3 are negligible, but strictly speaking the solution is so
far only an approximate one when the ends are free.
A New Method.
§ 5. In the Camb. Phil. Soc. Trans, vol. xv. pp. 313337
I showed how the mean values of the strains and stresses
might be obtained in any elastic solid problem independently
of a complete solution. For isotropic materials I obtained
(/. c. p. 318) three formulas of the type
E \\\ d ldxdydz= N[ {I*ri£Kw'+Yy))dx dy dz
■i;(Fff + (fy)}dS, . (4)
jj>
where a, fi, y are components of displacement, X, Y, Z of
bodily forces, and F, G, H of surface forces. The volume
integrals extend throughout the entire volume, and the surface
integrals over the entire surface of the solid. As was explicitly
stated in proving the results (I. c. p. 315), X, Y, Z may include
' reversed effective forces '
dPa d 2 /3 d 2 ry
p dt 2 ' p dF'  p W
where p is the density.
In the present application there are no real bodily forces ;
we may also leave surface forces out of account, if we suppose
that when one end of a rod is held, that end lies in the plane
2 = 0. F
Supposing the rod to vibrate with frequency &/27T, we have
so that we replace (4) by
"^JJJ ^ € *^^ €fe= ^^jHU ^^ — ^C«a?+ifiEy)l^»^«fc, . (5)
and similarly with the two other equations of the same type.
I he only other result required is one established in my
Vibrations in Solid and Hollow Cylinders. 337
paper (B) ; viz., that the general solution of the elastic solid
equations of motion in which the terms contain cos pz or
s'mpz consists of two independent parts. In the first, which
alone applies to longitudinal vibrations, a. and j3 are odd and
7 an even function of x and y.
We may thus assume
a = cos kt cos (pz — e) { A^ 4 A/y + A 3 # 3 4 A 3 'x*y
+a b 'v+a, ,, v +...},
/3=cos^cos(^.~e){B 1 A< + B/# + B 3 ^ 3 + ...}, I * W
7 = cos kt sin (pz  e) {C + C 2 x 2 + QJxy + C 2 ff f + •••}> )
where e is a constant depending on the position of the origin
of coordinates and the terminal conditions. It is obvious
from various considerations that the same e occurs in the
values of a, /3, y. Certain relations must subsist between the
constants A, B, C in the above expressions, in virtue of the
body stress equations, but we do not require to take any heed
of these for our present purpose.
§ 6. As the validity of solutions in series has been a subject
of contention in other elastic solid problems, some doubt may
be entertained as to the results (6). I would be the last to
deny the reasonableness of this, because I do not myself
regard (6) as universally applicable.
According to my investigations, quantities such as
(A 3 # 3 /A.i#) are of the order (greatest diameter/nodal length) 2 ,
and the series become less rapidly convergent as (greatest
diameter/nodal length) increases. In other words, increase
either in (greatest diameter/rod length) or in the order of the
" harmonic " of the fundamental note reduces the rapidity of
convergence. The proper interpretation, however, to put on
this is not that (6) is a wrong formula for longitudinal
vibrations, but simply that under the conditions specified the
vibrations tend to depart too widely from the longitudinal
type. If we apply this solution the results deduced from it
themselves tend to show the degree of rapidity of the con
vergence, and what we have to do is to keep our eye on the
results and accept them only so long as they are consistent
with rapid convergency.
Perhaps the following resume 1 of my views on this point
may be useful : —
1. In obtaining (6) originally I employed the complete
elastic solid equations for isotropic materials. In other cases
where difficulties have arisen over expansion in series, they
seem mainly due to the fact that the elastic solid equations have
been whittled down in the first instance for purposes of
338 JDr. C. Ohree on Longitudinal
simplicity. When one omits terms in a differential equation for
diplomatic reasons, the results may be perfectly satisfactory
under certain limitations. Owing, however, to the mutilation
of the differential equations, the resulting solutions are unlikely
to contain within themselves any satisfactory indication of the
limits to their usefulness. It is very much a case of running
a steamengine without a safetyvalve.
2. When the bar is of circular section and isotropic, the
series occurring in (6) are Bessel's functions of a wellknown
type, whose rapidity of convergence appears well ascertained
under the normal conditions of the problem. When the
section is circular, and the material not isotropic but sym
metrical round the axis, the series, whose mathematical law of
development I have obtained, converge to all appearance quite
as rapidly as in the case of isotropy. The other cases of
isotropic material — sections elliptical or rectangular — which
I have considered present similar features ; the only difference
being that the rate of convergence diminishes with increase
in any one dimension of the cross section.
3. If we suppose k = Q, or the vibrations to be of infinite
period, the solution must reduce to that for the equilibrium of
a rod under uniform longitudinal traction. ISTow, in the case
of equilibrium 7/0 reduces to a constant, while a and ft are
linear in x and y for all forms of cross section. The commencing
terms in series (6) are thus of the proper form under all
conditions, and the form of the differential equations shows
that if a, for instance, contains a term in 00 it must contain
terms in a? 8 , xy 2 , and other integral powers of % and y.
4. The general type of the differential equations is the
same for all kinds of elastic material, isotropic or ?eolotropic,
and the surface conditions are identical in all cases ; thus the
type of solution must always be the same. The results may
become enormously lengthy for complicated kinds of seolo
tropy, but by putting a variety of the elastic constants equal
to one another we must reduce the most complicated of these
expressions to coincidence with the corresponding results for
isotropy. Of course it does not follow that the convergence
will be equally rapid for all materials. A large value of a
Poisson's ratio in conjunction with an elongated dimension
of the cross section may reduce the convergency so much as
to throw the higher " harmonics " outside the pale of longi
tudinal vibrations.
5. The more the section departs from the circular form
the less rapid in general is the convergence, and the larger
the correction supplied by the second approximation. In
fact the size of the correction is probably the best criterion
Vibrations in Solid and Hollow Cylinders, 339
by which to judge of the limitations of our results. If the
correction is large even for the fundamental note, it is pretty
safe to conclude that the section is not one adapted for the
ordinary type of longitudinal vibrations. If a section, for
instance, were of an acutely stellate character, with a lot of
rays absent and the centroid external to the material, I for
one should be extremely chary of applying to it the ordinary
formula.
§ 7. For deiiniteness let us consider the fixedfree vibra
tions, taking the origin of coordinates at the centroid of the
fixed end. Our terminal conditions are
7 = when z = 0,
S=0 „ z = l;
the latter condition being the same thing as
_¥ =0 when z = L
dz
These conditions give at once
6=0, p=(2* + l)w/2Z,
and hence
I pz sin {pz — e) dz = 1 cos (pz — e) dz.
J Jo
Take the axes of x and y along the principal axes of the
cross section cr, so that
\\ xy dx dy = 0,
ij x 2 dx dy = k. 2 2 (t, Jj y 2 dx dy = k?<t.
Then, substituting from (6) in (5), we obtain at once
(EpFp/ / >)(C + Crf + C 2 / V + ...)
= ^(A 1 « 2 HB 1 V+...) . • (?)
The section is supposed small, L e. terms ^ in k{ 2 , k 2 2 are
small compared to O , though large compared to the terms
of orders /q 4 , &c, which are omitted in the above equation.
Thus as a first approximation the coefficient of C must vanish,
or
*=?VE/p, (1)
which is simply the ordinary frequency equation.
■* Treating the other two equations of the type (5) similarly,
340 Dr. 0. Cbree on Longitudinal
we obtain the two results
E(A 1 +3A 3 *^+A 3 'V + ...)
=  W 9 1 3 (C + C 2 * 2 2 + CV V + ...)+ 9B1V  W + . . .  , (8)
E(B 1 ' + B 3 V + 3B 3 'V+...)
_ k 9 P ^ (C + (W + C 2 % 2 + ...) + V Ai/c 2 2  B x V + . . . I . (9)
The terms not shown are of the order /q 4 , k 2 4 , or higher
powers of tc x and # 2 . Combining (8) and (9) we get
E(A 1 B 1 0=^p(l+7 ? )(A 1 ^ 2 B 1 V)+..., . . (10)
E (A, + B/) =  2k* P ( v >p) (C + C 2 * 2 2 + Co V + . . .)
+ (19,)^(A 1 / C2 2 +B 1 V) + ... • (11)
To see the significance of (10) replace k^p by its approximate
value /> 2 E , when we have
A 1 B/=(1 + ,){A 1 0« 2 )*B 1 >* 1 )*}+.. • (12)
As we have seen, p=(i + J)w/Z; and thus, so long as i is not
too large, {pfc 2 ) and (/^J 2 are in a thin rod small quantities
of the orders {kJI) 2 and (/cjl) 2 . Hence we deduce from (12)
as a first approximation
B/ = A ; (13)
This is all we require for our present purpose ; but, in pass
ing, it may be noted that as a second approximation we have
B^Mi+a+^Vi 2 */)}.
The more the section departs from circularity — i. e. the
more elongated it is in one direction — the greater is the dif
ference between B/ and A x . This and the fact that ik x jI and
IkJI must both remain small are useful indications of the
limitations implied in the method of solution.
Employing (13) in (7) and (11) we have, neglecting
smaller terms,
(BpAfy&O (Co + CW + CoV) = PprjA^ + K*),
2k*p Wp) (C + C 2 * 2 2 + C 2 V) =  2E Aj.
Whence we deduce at once, without knowing anything of
the constants C 2 , A 3 , &c,
Wpk* P /p)+(2P(»,/p) =t?M<h* + *i*)/(2E).
Vibrations in Solid and Hollow Cylinders. 341
Using in the small term (that containing & 4 ) the first ap
proximation (1), we have
F /3 =E^{i^ 2 (« l 2 +«/)}; • • • (14)
and this, as
simply reduces to (3).
That the above proof is as satisfactory in every way as one
based on ordinary elastic solid methods I should hesitate to
maintain. Unless one knew beforehand a good deal about
the problem there would, I fear, be considerable risk of mis
adventure.
§ 8. In illustrating the method in detail I have selected
the case of isotropic material simply because I did not wish
to frighten my readers. The assumption of isotropy almost
invariably shortens the mathematical expressions, and gene
rally also simplifies the character of the mathematical opera
tions ; and isotropic solids thus flourish in the textbooks to a
much greater degree than they do in nature. When, how
ever, the mathematical difficulties are trifling, as in the
present case, it seems worth while considering some less
specialized material. I shall thus briefly indicate the appli
cation of the method to the case of material symmetrical with
respect to the three rectangular planes of x, y, z, taken as in
the previous example. In this case the stressstrain relations
involve, on the usual hypothesis, nine elastic constants.
Such quantities as Young's modulus or Poisson's ratio must
be defined by reference to directions. Thus let E x , E 2 , E 3
denote the three principal Young's moduli, the directions I, 2, 3
being taken along the axes of x, y, z respectively. There
are six corresponding Poisson's ratios, each being defined by
two suffixes, the first indicating the direction of the longitu
dinal pull, the second that of the contraction. For instance,
rj 12 applies in the case of the contraction parallel to the
yaxis due to pull parallel to the #axis. The order of the
suffixes is not immaterial, but there exist the following
relations : —
WEi=W E »; WEj=WEi.J W E 3 = W*V (15)
The three equations answering to (4) are
vjj fo dz dy dz= k * p Jj] frsto^^y}** *9 d ~, ( 16)
Ei n)s^^^ = ^]Ij ^ ax ~ ri ^~" ni ^ dxd y dz ^ i 17 )
342 Dr. 0. Chree on Longitudinal
From the nature of the elastic solid equations the expres
sions for the displacements must be of the same general form
as for isotropy, so that we may still apply the formulae (6)
for a, ft, y. Doing so, and following exactly the same pro
cedure as in the case of isotropy, we obtain from (16), for
any shape of section,
(E 3 p  k*plp) (C + ...) =  *V(%i A^ 2 + *AV) + • . , (19)
and from (17) and (18) as first approximations
A 1 /(%3E 3 /E 1 )=B 1 '/( %3 E 3 /E 2 ) = p(C +...). . (20)
Thence we obtain at once
k*p=r>E 3 {lp*JZ 3 (^W+ ?gV)} • (21)
Employing (15) we give this the more elegant form
*V j*E»{ l f ( Vsi W + % 2 V) \ ,
whence
k=p(K 3 / P )Hli P *(vnW + V S 2W)\. ■ (22)
For a circular section of radius a
k=p(E i /p)i{l±p*a*( V51 * + V32 *)}. . . (23)
For a rectangular section 2a x 2b ; the side 2a being parallel
to the #axis,
k=p(E 3 /p)h{lip*( a * V3] ? + b\ 3 /)\. . . (24)
For a given size and shape of rectangle, the correction to
the first approximation is largest when the longer side is that
answering to the larger Poisson's ratio (for traction along
the rod). Possibly experimental use might be made of this
result in examining materials for seolotropy.
If the material, though not isotropic, be symmetrical in
structure round lines parallel to the length of the rod,
V3 i =Vs2 = V, say,
and writing E for E 3 in (22) we reproduce the result (60) of
paper (C).
The results (22), (23), and (24), so far as my knowledge
goes, are absolutely new.
Extension of Earlier Results.
§ 9. My paper (A), like the corresponding investigation of
Pochhammer, dealt only with a solid circular cylinder ; but
the same method is applicable to a hollow circular cylinder.
For greater continuity I shall employ in the remainder of
tbis paper the notation of paper (A) .
Vibrations in Solid and Hollow Cylinders. 343
The displacements are u outwards along r, the perpendi
cular on the axis, and w parallel to the axis, taken as that of
z. Thomson and Tait's notation 711, n for the elastic constants
is employed. TJie frequency is k/2w and the density p, as in
the earlier part of this paper, and for brevity
&p/(m + n)=J, k 9 p/n = P, . . . (25)
so that a. and ft have utterly different significations from
their previous ones.
There being no displacement perpendicular to r, in a trans
verse section, the dilatation 8 is given by
5. du u dio , x
S= A + r + dz ^
It was shown in paper (A) that the following equations held
#8 .ldB. d*8
d? + v<h + d;* +aB=0 > • • • • ( 2? )
*£ + !£_« +£+,<*,= _*<» (28 )
dr* r dr r 2 dz l n dr K '
d 2 w , 1 div , d 2 w g m d8 ,_.
f/r* 2 ?' dr a* 2 rc dz
Employing J and Y to represent the two solutions of
BessePs equation we find, as in paper (A), that the above
equations are satisfied by
8 = cos kt cos (pz  e) { G J {r (a 2  f)h) + 0' Y (K" 2 p 2 )h) } , (30)
11= cos ft cos {pze) [AJ\{r(/3 2 p 2 )i) + A'Y l (r{/3 2 p^)
~f i iE^\ GJ M^P 2 ) i ) +O r Y 1 (r(««^*)}], (31)
mj= cos fa sin Cp*e) R AJ (?'(/3 2 ^ 2 )*) + A'Y (r(/3 2 p 2 )t) j
x (fag + ^j*^ {OJo(Ka»^)») + PYo(r(a»^)i)}] > (32)
where A, A', C, C are arbitrary constants to be determined
by the surface conditions.
In reality ofip 2 is negative ; but the properties of the
J and Y Bessel functions which at present concern us
are not affected thereby. ft 2 —p 2 , on the other hand, is
positive.
344 Dr. C. Chree on Longitudinal
§ 10. If a and b are the radii of the outer and inner cylin
drical surfaces respectively, then from the conditions which
hold over these surfaces we must have
*+%*> ™
(mn)B + 2np=0, .... (34)
when r = a, and when r=b.
As regards the terminal conditions we should have, follow
ing the ordinary view of longitudinal vibrations,
M? = over a fixed end, .... (35)
a free end. (36)
zz = (m — n)$\2n—=0,
> over a
^(S+f)=° *
We have no means of satisfying these terminal equations
by means of the present solution save by selecting suitable
values for p and e. Clearly if both ends e = and z = l be
fixed we accomplish our object by putting
e = 0, p = iirjl.
If, however, z = l be a free end, while ^ = is a fixed, we
must have e = to satisfy the conditions of the fixed end ;
and this leaves us with
zz cc cos pi,
zr oc sin/>Z
over the free end. This is the difficulty we have already
indicated in § 4 ; and it is in no respect peculiar to hollow
cylinders, and need not further concern us at present.
§11. In dealing with the surface conditions, brevity is
effected by the use of the notation
\=>v/a 2 p 2 , /xeee V73 2 ;? 2 , . . (37)
whence
a 2 _/3 2 = \ 2 Ar.
After simplifications, into which I need not enter, the elimi
nation of A, A', C, C from the four equations holding over
the cylindrical surface supplies the determinantal equation
Vibrations in Solid and Hollow Cylinders.
345
y,(*m)
2p 2 A
J,(«X)
2p 2 X
a (^ — i> )
m — n \ 2 T ,
Y^aA)
a 2 (^ 2 l' 2 )
2x> 2 X
%_ X 2
2nfx
m— n
Y,'(6X)
h — w. X 2
2n/u oV J a 2 //
This equation is true irrespective of the relative magnitudes
of a and b. It constitutes a frequency equation supplying
values of k which apply to the type of vibrations consistent
with the surface conditions. If both ends of the rod be fixed
there is no restriction to the absolute values of a/l and b/l;
but if one or both ends are free, such a restriction is really
involved in the fact that unless ia/l and ib/l be both small —
i being the order of the harmonic of the fundamental note
under consideration— the failure to satisfy exactly the terminal
condition ^
zr —
involves an inconsistency which cannot be allowed.
§ 12. The case of a thick rod fixed at both ends is of little
physical interest, and the treatment of (38) in its utmost
generality would involve grave mathematical difficulties.
I thus limit my attention to the case when ia/l and ib/l are
both small. This implies that a\, b\ ayb i and b/n are all small.
Thus in dealing with the various Bessel functions we may use
the following approximations*, which hold so long as the
variable x is small,
J (#) = 1^/4,
^(^ = 1^(1 <z> 2 /8),
Y (*) = (1^/4) log tf + 074, ^ ' ( 39 )
:0.
(38)
YiW = (l
x 2 / '8) log x— # _1 — w/4tj
Y/(a ? )=i(l^)log^ + ^ 2 + . J
Ketaining only the principal terms in (38), we are of course
led at once to the first approximation (1). Again, if (1) held
exactly w T e should find
« 2 =p 2 n (3m — n) f m (m + n) ,
fju 2 — p 2 =p 2 (m — n) /m ;
* Cf. Gray and Mathews' ' Treatise on Bessel Functions,' pp. 11, 22, &c.
'•}
346 Dr. C. Chr'ee on Longitudinal
and to a first approximation
J\ (aX) 2p 2 X 2p 2 X 2 _2 m(mn)
Ji(ajj,) a\fjb 2 —p 2 ) <z 2 fi(fjb 2 — p 2 ) fi n{3m—n) 9
m—n J (aX) X 2 J/(aX) _ 2(m— n) X 2 _ 2m(m — n)
2nfjb Ji(a/ji) a 2 /n J / (aft) 2n/j, afi fin^dm — nf
and similarly if a be replaced by b.
We thus see that the third column in the determinant (38)
is such that each principal term in it is obtained by multi
plying the principal term in the same row in the first column by
the same constant 2m (m ~ n)h \/jbn(3m — n) \. Now if one
column of a determinant is obtainable by multiplying another
column by a constant that determinant vanishes. It is thus
at once clear that in proceeding even to a second approxima
tion we need retain only the principal terms in the second
and fourth columns of (38). This removes what seemed at
first sight a formidable obstacle, viz. the occurrence of the
logs in the expressions for Y , Y x , and Y/.
§ 13. For further simplification of the determinant multiply
each term by 2, and divide the first and second rows by a/n
and bfi respectively. Then for the second row write the
difference between the first and second rows, and for the
fourth row the difference between the third and fourth rows,
and multiply the resulting rows by b 2 /(a 2 — b 2 ). Finally
multiply the second column by a 2 fi 2 /2, the third column by
[Aa?, and the fourth column by a 2 fji<z 2 /2. We thus reduce (38)
to the easily manageable form
iia,y l _^>! 2 (i_i^) M,
[M Z —p~ K ° ' fl 2 —p 2
xyb 2 2 P 2
1 A2..2 1
8 T H^p 2 ) fi 2 p 2
1faV 1 x» + ^^^(^ + W , ?=5) 1
%hY 1 ib 2 X 2 (3X 2 + 2a 2 ^) 1
After algebraic reduction, use being made in the secondary
terms of the first approximation results (1), (40), &c, we
easily deduce from (41)
l
(4]
whence
»**^{iv(*£](!*+»>}. («>
l=p(E/p)i{l^^rf a ±ty . . . (43)
Vibrations in Solid and Hollcw Cylinders. 347
In a hollow circular section the radius of gyration round
the perpendicular to the plane through the centre is given by
K 2 =(a 2 + b*)/2,
so that (43) is in agreement with (3) and (22).
The fact that (43) is merely a special case of (3) or (22)
may seem to indicate that our separate treatment of the
hollow cylinder, or tube, is quite unnecessary. I can only
say that having regard to the methods by which (3) and (22)
were arrived at — more especially to the fact that in establishing
(6) I was dealing with solid cylinders — I had long felt the
desirability of an independent investigation.
§ 14. The complete determination of the constants A, A 7 ,
C, C^, and of the several displacements, strains, and stresses
to the degree of accuracy assumed in (43), though not a very
arduous labour, would require more time than seems warranted
by the physical interest of the problem. I thus confine my
further remarks to the form of the longitudinal displacement
w. Substituting their approximate values for tbe J's and
Y's from (39) in (32), we find
piDJfJb cos M sin (pz — e) =
 A(l  i/zV) ~ A'{ (1  i*V) log fir + JpV \
Now, considering only their principal terms, it is easily
seen that A'/ A and C/C are both of the order (p 2 ab) 2 . Thus,
to the present degree of approximation, we may leave the A'
and G / terms in iv out of account. Also confining ourselves
to principal terms, we easily find
A = Cp 2 /fJLCC 2
2(m—n) ~ m + n
Hence, employing the two last of equations (40), we deduce
i{ Afi 2  (Cp 2 /y.a 2 )\ 2 \ H {  A + Cp 2 /f*« 2 }
We thus have from (44), to the degree of approximation
reached in (43),
iv = iv cos kt sin (pz — e) (1 — i^pV) , . . (45)
where iv is a constant which depends on the amplitude of the
vibration.
The expression (45) for iv is exactly the same as I found
348 Longitudinal Vibrations in Solid and Hollow Cylinders.
in my earlier papers for solid cylinders ; the r 2 term repre
senting one of the additions I deem it necessary to make to
Lord Ray lei gh's assumed type of vibration.
The paraboloidal form of cross section, met with except at
nodes or when cos kt vanishes, seems to me an interesting
feature of the longitudinal type of vibrations. Possibly,
observations on light reflected from a polished terminal face
might lead a skilled experimentalist to interesting conclusions
as to the value of n.
It should, however, be borne in mind that, inasmuch as the
terminal condition zr = is not exactly satisfied by the above
solution in fixedfree vibrations, there may be a slight departure
from the theoretical form in the immediate neighbourhood of
a free end.
§15. The result (43) is true irrespective of the relative
magnitudes of b and a. If h/a be very small, the correctional
term is the same as for a solid cylinder of the same external
radius. If, on the other hand, b/a be very nearly unity, or
the cylinder take the form of a thin walled tube, we have
Jc=p(E/p)HlhP*vW) (46)
The correctional term is here twice as great as in a solid
cvlinder of radius a.
* § 16. An experimental investigation into the influence of
the shape and dimensions of the crosssection on the frequency
of longitudinal vibrations is certainly desirable. In com
paring the results of such an investigation with the theoretical
results here determined, several considerations must, however,
be borne in mind.
Statical and dynamical elastic moduli are to some extent
different, so that the value of E occurring in (2) or (3) is not
that directly measured by statical experiments*. In other
words the difference between the observed frequency of the
fundamental vibration in a fixedfree bar, and the frequency
calculated from the ordinary formula
if E be determined directly by statical experiments, is not to
be wholly attributed to the defect of the ordinary first
approximation formula. Again, it must be remembered that
E varies t, often to a very considerable extent, in material
nominally the same ; so that the difference of pitch observed
* See Lord Kelvin's Encyclopaedia article on Elasticity, § 75, or
Todhunter and Pearson's ' History,' vol. iii. art. 1751.
t For the effects of possible variation in the material throughout the
bar, see the Phil. Mag. for Feb. 1886, pp. 81100.
Experiments on Artificial Mirages and Tornadoes. 349
in different rods cannot without further investigation be safely
ascribed to differences in the area or shape of their cross
sections. Further, elastic moduli may alter under mechanical
treatment, so that it would be unsafe to assume if a hollow
bnr were further hollowed or were altered in shape, that its
Young's modulus would remain unaffected.
If it were possible to measure with sufficient accuracy the
frequency of the fundamental note and several of its
" harmonics " in a single rod, one would have a more certain
basis of comparison with the theoretical results. Even in
this case, however, there is the consideration that in practice
the rod must be supported in some way, and this is likely to
introduce some constraint not accurately represented by the
theoretical conditions. Again, reaction between the vibrating
rod and the surrounding medium may not be absolutely
without influence on the pitch *.
I mention these difficulties because their recognition may
prevent a considerable waste of time on the part of anyone
engaged in experiments on the subject.
Though somewhat of a side issue, it may be worth remark
ing that the correction factor l — ^p^ific 2 for the frequency in
isotropic material contains no elastic constant except Poisson's
ratio. Thus observations made on rods differing only in
material might throw some light on the historic question
whether rj is or is not the same for all isotropic substances.
The discussion of equations (6) and of the experimental
side of the problem has been largely expanded at the suggestion
of the Society's referee.
XXX. Some Experiments on Artificial Mirages and
Tornadoes. By R. W. Wood f.
[Plate III.]
IN an article published in i Nature ' for Nov. 19, 1874,
Prof. Everett, in discussing the phenomenon of mirage,
showed that the condition necessary for the formation of sharp
images in a horizontally stratified atmosphere, is a plane of
maximum refractive index, the optical density decreasing as
we go above or below this plane in direct proportion to the
distance.
A horizontal or nearly horizontal ray will be bent towards
and cross the plane of maximum density, where it changes
its curvature and is again bent towards the plane, which it
* Cf. Lamb, Memoirs and Proceedings Manchester Phil. Society,
vol. xlii. part iii. 1898.
f Communicated by the Author.
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 B
350 Mr. R. W. Wood on some Experiments
may thus cross again and again, traversing a path which is
approximately a sine curve.
While showing the curvetrajectory of a ray of light in a
vessel filled with brine, the density of which increased with
the depth, it occurred to me that by properly regulating the
refractive index of the liquid, the ray might be made to
traverse a sine curve.
Some attempts in this direction were so successful, and
yielded such beautiful experiments for the lectureroom, that
it seems worth while to publish them, together with some
photographs of the trajectories, although, as I have since
learned, very similar experiments have already been described
by Wiener ( u Gekriimmte Lichtstrahlen," Wied. Annalen,
xlix. p. 105).
For the liquid 1 adopted an arrangement very similar to
the one described by Prof. Everett for obtaining mirage in a
rectangular tank.
A trough was first made of plate glass, about 50 cms. long,
10 cms. bigh, and 2 cms. wide. This was filled to the depth
of 3 cms. with a concentrated solution of alum. By means of
a pipette, of the form shown in PI. III. fig. 1, water containing
about 10 per cent, of alcohol was carefully deposited on the
alum solution to a depth of 3 cms. The addition of the alcohol
brings up the refractive index of the water, and is necessary
for reasons that will be spoken of presently. As a liquid of
high index, with a specific gravity intermediate between that
of the other two liquids, I used, instead of sugar and whiskey,
a mixture of glycerine and 85 per cent, alcohol, the right pro
portions being easily found by experiment.
The mixture should float on the alum solution and sink in
the water, and is introduced between the two layers by means
of the pipette at the end of the trough through which the ray
is to enter. A layer about 3 cms. in thickness will be found
about right.
All three of the solutions should be first acidified with a few
drops of sulphuric acid, and rendered fluorescent with quinine.
The difference in surfacetension between the two upper
layers may give some trouble: when the pipette is withdrawn
it may draw a thread of the glycerine and alcohol mixture up
through the water, and a complete upsetting of the layers
occur as a result of the forces of surfacetension.
This invariably happens when alcohol is not added to the
water, and can be remedied either by the addition of water to
the glycerine mixture or of more alcohol to the water. It is
a good plan in withdrawing the pipette to pull it out slowly
in a very oblique direction, in order that the heavy liquid may
be washed off before the tube reaches the surface.
on Artificial Mirages and Tornadoes. 351
The three layers may now be cautiously stirred to hasten
the diffusion, after which they should be allowed to rest a few
minutes until the strias have disappeared.
If now a beam of light from an arclamp, made parallel by
means of a condensinglens, be thrown obliquely into one end
of the trough, it will be seen to traverse the liquid in the
form of a most beautiful blue wave, the curvature of which
varies with the angle at which the ray enters. Rays of
light travelling in sine curves are shown in figs. 2 and 3,
which were photographed directly from the trough.
Prof. Everett showed in his paper that a parallel or slightly
divergent ray entering a medium of this description would,
converge to a linear focus, and then successively diverge
and converge to conjugate foci. This phenomenon is shown
in fig. 4, which is a photograph of the trough with a rather
wide beam of horizontal parallel light entering the end.
This experiment I have never seen described, though Exner
has shown that the eyes of some insects operate in a similar
manner, the visual organ consisting of a transparent cylin
drical body, the axis of which has a high refractive index,
while as we approach the surface the optical density decreases
continuously.
The beautiful miniature desertmirages that I have witnessed
on the level city pavements of San Francisco (see letter and
photograph in ' Nature ' for Oct. 20, 1898), suggested to me
the idea of reproducing this phenomenon on a small scale in
the classroom.
Although 1 have already described very briefly an expe
riment of this nature, I will repeat the description now some
what more in detail. Three or four perfectly flat metal
plates, each one about a metre long and 30 cms. wide, should
be mounted end to end on iron tripods and accurately levelled.
The plates should be thick enough not to buckle when heated,
say 0'5 cm. I have used plaster plates, made by casting
plaster of Paris on plate glass, with some success, though they
are fragile and not very durable.
Probably plates of slate would serve admirably, since they
will stand a fair amount of heating, and can be obtained very
flat and smooth.
The plates must be thickly sprinkled with sand to destroy
all traces of reflexion at grazing incidence, and the sand sur
face should appear perfectly level when looked along from a
point just above its plane. On the absolute flatness of the
desert depends the successful working of the experiment;
therefore too much care cannot be given to the adjustment of
the plates. An artificial sky must be formed at one end of the
desert. If the experiment is being performed at night, a
2 B2
352 Experiments on Artificial Mirages and Tornadoes.
sheet of thin writingpaper with an arclamp behind it works
very well, but a large mirror set in a window and reflecting
the sky is better, when daylight is to be had.
Between the sky and the desert a small range of mountains,
cut out of pasteboard, should be set up. The individual peaks
should be from 1 to 2 cm. high, and the valleys between them
should be only a trifle above the level of the desert. The general
arrangement is shown in fig. 5.
The plates are now heated by means of a row of burners,
which should be moved about from time to time in order to
prevent overheating any one place.
If now we look along the desert, holding the eye only a
trifle above the level of the sand, we shall see the mountains
sharply outlined against the sky: as the temperature rises a
lake begins to form in front of the mountainchain, and in a
few moments the inverted images of the peaks appear as if
reflected in the water. If the eye be depressed a trifle, the
base of the mountainchain vanishes completely in the illusory
lake, which now appears as an inundation. These appear
ances are shown in fig. 6, the photographs having been taken
of the actual mirage on the artificial desert. The first of the
three shows the appearance when the plates are cold, the
second the apparent lake with the images of the peaks in the
water, and the third the vanishing of the lower portions of
the range. Two or three palmtrees, cut out of paper, were
stuck up to add to the effect. Vertical magnification can also
be shown on the hot desert: if the mountains are removed
and a small marble be laid on the sand at the farther end of
the desert, it will be found that if the eye be brought into
the right position, the circular outline will change into an
ellipse, and as the eye is lowered the image will contract to a
point and eventually disappear. The magnification in this
case is of course due to the running together of the direct
and refracted images. I have observed similar cases in
looking across our lake, when the water was warm and the
air cold, patches of snow on the opposite shore, too small to
be visible to an eye several metres above the level of the lake,
coming out very distinctly when one walked down a bank to
the water's edge.
The atmospheric conditions existing when mirages of this
description are observed are such as give rise to the dust
whirls, so often seen on the American desert, and when
existing on a larger scale, to tornadoes. There seemed no
reason why these whirlwinds should not be produced on a
small scale as well as the mirages. One of the metal plates
was sprinkled with precipitated silica and heated with a few
burners : in a few minutes most beautiful little whirlwinds
On the Thermal Properties of Normal Pentane. 353
began to run about over the surface, spinning the fine powder
up in funnelshaped vortices, which lasted sometimes ten or
fifteen seconds. The silica powder must be made by igniting
the gelatinous precipitate formed when silicon tetrafluoride is
conducted into water. The commercial article is not suffi
ciently light and mobile.
Whirls formed in this way cannot be seen by a large
audience, however, and I accordingly sought some way of
making them on a larger scale. The plate was well heated
after removing the silica and then dusted with salammoniac:
dense clouds of white vapour immediately arose from the hot
surface, and presently in the centre there mounted to a height
of about 2 metres a most perfect miniature tornado of dense
smoke. By placing the plate in the beam of a lantern in a
dark room, the whirls can be shown to a class in a large
lectureroom. I find that it is best to put on the salammoniac
first and then heat the plate: the vortices then come off the
plate almost continuously, and often persist for some time.
An instantaneous photograph of one of these tornadoes was
taken in bright sunshine, and is reproduced in fig. 7.
This method of showing atmospheric vortices seems far
preferable to the old way of forming them, by means of a
rapidly whirling drum with cross partitions, as the whirls are
produced by the same causes and under the same conditions
that they are in nature.
Physical Laboratory
of the University of "Wisconsin,
Madison, Nov. 20.
XXXI. On the Thermal Properties of Normal Pentane. By
J. KoseInnes, M.A., B.Sc, and Sydney Young, D.Sc,
F.R.S*
IX the year 1894 an experimental investigation of the
relations between the temperatures, pressures, and
volumes of Isopentane, through a very wide range of volume,
was carried out by one of us, and the results were published
in the Proc. Phys. Soc. xiii. pp. 602—657. It was there shown
that the relation p = bT — a at constant volume (where a and
b are constants depending on the nature of the substance and
on the volume) holds good with at any rate but small error
from the largest volume (4000 cub. cms. per gram) to the
smallest (1*58 cub. cms. per gram).
In the neighbourhood of the critical volume (4*266 cub.
cms.), and at large and very small volumes, the observed
deviations were well within the limits of experimental error
* Communicated by the Physical Society: read December 9, 1898.
354
Mr. J. RoseInnes and Dr. S. Young on th
te
but at intermediate volumes they were somewhat greater, and,
as they exhibited considerable regularity, it is a question
whether they could be attributed entirely to errors of experi
ment. In any case, the relation may be accepted as a close
approximation to the truth.
A quantity of pure normal pentane having been obtained
by the fractional distillation of the light distillate from
American petroleum, it was decided to carry out a similar
investigation with this substance; but, as it had been found
that isopentane vapour at the largest volumes behaves
practically as a normal gas, it was not considered necessary
to make the determinations through so wide a range of
volume.
The method employed for the separation of the normal
pentane from petroleum has been fully described in the Trans.
Chem. Soc. 1897, Ixxi. p. 442 ; and the vapourpressures,
specific volumes as liquid and saturated vapour, and critical
constants have been given in the same journal (p. 44G).
The data for the isothermals of normal pentane were
obtained by precisely the same experimental methods as in
the case of isopentane, and reference need, therefore, only be
made to the previous paper (loc. cit.) .
There were four series of determinations ; and particulars as
to the mass of pentane, and the data obtained in each series
are given below : —
Series.
Mass of
[Xormal Pentane.
Data obtained.
I.
11.
III.
i IV.
gram.
•10922
•02294
•005858
•001845
Volumes of liquid to critical point ; volumes above
critical temperature to 280°.
Volumes of unsaturated vapour from 140° to critical
point ; volumes above critical point to 280°.
Volumes of vapour at and above 80°.
Volumes of vapour at and above 40°.
The correction for the vapourpressure of mercury was
made in the same way as with isopentane : when liquid was
present it was assumed that the mercury vapour exerted no
pressure ; in Series I., above the critical point, onefourth
of the maximum vapourpressure of mercury was subtracted;
in Series IT. onehalf ; in Series III. threefourths ; and in
Series IY. the full pressure.
The volumes of a gram of liquid and unsaturated vapour
are given in the following table.
Thermal Properties of Normal Pentane, 355
Volumes of a Gram of Liquid and of Unsaturated Vapour.
Series
1.
rp j Pressure.
1 ' millim.
Volume, rn i
cub. cm. P"
Pressure.
Volume.
Temp.
Pressure.
V lume.
millim.
cub. cm.
millim.
c u . cm
130°. 7023*
2022 170°.
29270
2269
200°.
26910
3329
8104
2018 (cont)
31990
2250
(cont.)
27670
3136
11430
2008
35040
2230
29350
2944
15100
1998
38270
2211
30820
2847
1881
1989
42170
2191
33060
2751
22630
1979
46180
2172
36260
2655
26810
1969
50760
2154
41090
2558
31010
1959
55660
2135
4(5560
2482
; 35560
1950
180°.
19340
2580
49980
2443
40280
T940
20110
2557
51920
2424
45520
1930
20770
2538
54060
2405
50450
1921
21570
2519
210°.
28410
6055
140°.
8507*
2094
22470
2500
28790
5666
9927
2089
23420
2481
29180
5274
10580
2 085
24570
2462
29530
4883
13230
2075
25730
2443
29920
4492
15730
2066
27200
2423
30400
4 106
18280
2 056
28550
2404
31160
3716
21300
2046
30230
2384
32660
3330
24140
2036
32060
2365
34200
3137
27300
2027
33970
2346
36870
2945
30670
2018
36240
2327
39010
2848
34610
2008
38630
2307
41970
2752
38340
1999
41100
2288
46070
2656
42710
1989
44280
2269
49360
2598
46880
1979
47430
2250
51950
2559
150°.
11380*
2172
50830
2231
54940
2520
14680
2153
54400
2211
220°.
31010
6056
18770
2133
190°.
22490
2900
31610
5667
22830
2114
22750
2866
32270
5275
27840
2095
23180
2827
32920
4884
33180
2075
23690
2789
33670
4493
39100
2056
24310
2750
34630
4107
45700
2037
25110
2712
36030
3717
53080
2018
261C0
2673
38540
3331
160°.
14060
2272
27230
2635
40790
3138
14330
2268
28630
2596
44510
2946
16670
2 249
30340
2558
47320
2849
19250
2229
32340
2520
51110
2752
22000
2210
34780
2481
56010
2657
25350
2491
37650
2443
230°.
33630
6058
29010
2172
41060
2404
34120
5669
32900
2153
43110
2385
35320
5277
37460
2134
45310
2366
36320
4885
42200
2115
47620
2346
37460
4495

47860
2095
50200
2327
38930
4108
53780
2076
52950
2308
40960
3718
170°.
16560
2398
200°.
25720
6053
42530
3525
17520
2384
25890
5665
44530
3331
19020
2364
26030
5273
1
47610
3139
20660
2345
26100
4881
49680
3042
22520
2326
26190
4491
52410
2946
24460
2307
26250
4105
55850
2850
26630
2288
26400
3715
240°.
36240
6059
Pressure below vapourpressure.
356 Mr. J. RoseInnes and Dr. S. Young on the
Series I. (continued).
Temp.
Pressure.
Volume.
Temp.
Pressure.
Volume.
Temp.
Pressure.
Volume.
millim.
cub. cm.
millim.
cub. cm.
millim.
cub. cm.
240°,
37310
5670
250°.
45160
4497
270°.
43780
6064
(cont.)
38430
5278
(cont.)
47720
4110
45570
5675
39770
4886
49390
3915
47600
5282
41350
4496
51370
3720
49990
4890
43310
4109
53890
3527
51320
4695
46170
3719
260°.
41260
6062
52890
4499
48150
3526
42850
5673
54600
4306
50870
3332
44590
5281
! 280°.
46310
6065
54160
3140
1
46530
4889
48350
5676
250°.
38810
6 061
49020
; 4498
50620
5283
40110
5672
50420
4304
51960
5087
41510
5279
52190
4111
53300
4891
43230
4888
1
54090
3916
II
54930
56810
4696
4500
Series
II.
Pressure.
Volume.
I
Temp.
Pressure.
Volume.
Temp.
Pressure
' Volume.
Temp.
millim.
cub. cm.
millim.
cub. cm.
millim.
cub. cm.
140°.
8866
3061
200°.
18460
1493
240°.
14570
2513
9055
2970
(cont.)
19160
1401
(co?it.)
15530
2327
9255
2877
19880
1310
16600
2141
9466
2785
20660
1218
17810
1956 i
9682
26 93
21470
1127
19220
1771
9886
2600
1 22300
1035
1
20380
1632 I
160°.
9822
2971
 23130
944
21730
14 95
10570
2694
23950
852
22710
1403
11100
2508
24720
7 61
23750
1311
11700
2322
25330
669
24910
1219
12340
2136
25620
6 23
26160
1128
13010
1952
220°.
11960
2976
27560
1036
13420
1860
12930
2698
29070
945 i
13790
1767
13720
2511
30700
853 I
180°.
10550
2973
14580
2325
32570
761
11380
2696
15560
2139
34670
670
12000
2509
16660
1955
35790
624
12690
2323
17900
1770
36990
578
13450
2137
18950
1631 1
38330
532
14280
1953
20100
1494
39830
487
15200
17 68
20950
1402
41730
441
15700
1676
21850
1310
260°.
13340
2979
16220
1584
22810
1219
14500
27 01
16770
1493
23850
1127
15410
2514
17320
1401
24950
1035
16440
2328
17890
1309
26140
944
17610
2142
18460
1217
27380
852
18940
19 57
19010
1126
28720
761
20480
1772
200°.
11260
2975
30080
669
21810
1633
12160
2697
30720
623
23300
1495
12870
2510
31440
578
24410
1404
13640
23 24
32160
532
25620
1312
14510
2138
32990
486
26960
1220
15470
1954
33900
440
28460
1128
16570
1769
240°.
12650
2977
30090
1036
17480
1630
13740
2700
31900
945
Thermal Properties of Normal Pentane.
Series IT. (continued).
357
„ Pressure,
Volume
Temp.
Pressure
Volume.
Temp.
Pressure.
Volume.
Tem P millim.
cub. cm
millim.
cub. cm.
millim.
cub. cm.
260°. 33970
853
280°.
16210
2515
280°.
30600
1129
(cont.) 36340
762
(cont.)
17320
2329
(cont) i 32550
10 37
39120
670
18580
2143
34680
946
40630
624
20030
1958
37130
854
42360
5 78
21710
1773
39980
762
41370
532

23160
1634
43400
6 70
46(540
487
24810
1496
45350
624
49650
441
26050
1405
47680
579
280°. i 13980
2980
27410
I312
50280
532
: 15230
2702
28940
1221
52150
54120
505
478
Series III.
T I
Pressure.
Volume.
Temp.
Pressure
Volume.
r n Pressure.
Volume.
Temp.
millim.
2371
cub. cm.
11611
millim.
cub. cm.
3332
lemp.
millim.
cub. cm .
80°. '
140°
8325
200°
13140
2440
2441
11250
(cont.)
9040
2974
(cont.)' 13930
2262
2510
10890
9435
2795
240°. 3631
11659
2586
10529 :
9860
2616
3988
10572
2666
10164
160°.
3015
11635
4276
9840
2708
9983
3301
10550
4603
9110
100°.
2537
11617
3532
9820
4980
8383
2770
10534
3795
9091
5428
7661
2955
9805
4092
8365
5958
6934
■
3166
9077
4448
7645
6426
6391
3405
8352
4859
6920
6970
5854
3681
7633
5231
6378
7372
5495
4006
6909
5652
5842
7823
5134
4190
6550
5979
5483
8362
4775
4398
6190
6324
5123
8965
4416
120°.
2701
11623
6733
4765
9677
4057
2951
10539
7192
44 07
10500
3700
3152
9810
7685
4049
11480
3341
3376
9082
8284
3692
12650
2982
3638
8356
8977
3334
13330
2802
3940
7637
9795
29 76
14090
2623
4294
6913
10260
2797
14920
2442
4607
6371
10770
2617
15870
2264
4960
5836
11310
2437
280° 3933
11671
5234
5478
11890
2259
4337
105 83
5531
5118
200°.
3317
11647
4656
9850
5859
4760
3642
10561
5022
9119
6233
4402
3896
9830
5440
8392
6639
4044
4190
9101
5936
7668
140°.
2861
11629
4533
8374
6511
6941
3129
10545
4927
7653
7024
6398
3342
9815
5401
6927
7634
58 60
3587
9087
5812
6385
8074
5500
3870
8360
6299
5848
8596
5139
4197
7641
6665
5489
9187
4780
4577
6916
7084
5129
9878
4420
4920
6375
7554
4770
10670
4061
5316
5839
8066
4412
11600
37 04
5607
5480
8678
4053
12700
3344
5943
5121
1
9387
3696
14050
2985
6309
47 63
10240
33'37
14830
2805
6717
4405
11240
2979
15690
2625
7179
4047
11810
2799
16(580
2445
7708
 3690
12440
2(520
!
17800
2266
358 Mr. J. KoseInnes and Dr. S. Young on the
Series IV.
(Temp.
J
Pressure.
Volume.
Temp.
Pressure.
Volume.
Temp.
Pressure.
Volume.
millim.
cub. cm.
millim.
cub. cm.
millim.
cub. cm.
40°.
858
2993
120°.
928
3576
200°
2415
1628
869
2935
989
3346
(cont.)
2588
1515 !
I 60 f '.
857
322 6
1061
31V5
2788
1401
1
889
3110
1143
2884
3024
1287
J
956
2879
1237
2653
3302
1174
1034
2649
1350
2425
3458
1116
1125
2421
1485
2195
3632
1060
1236
2191
1604
2023
3830
1002
1299
2077
1745
1853
4049
946 i
1370
1963
1854
1739
240°.
1217
3587 i
1447
185
1977
1625
1299
3357
1534
1737
2116
1511
1394
3124
1580
168
2276
1398
1502
289 3 ,
80°.
882
3343
2463
1284
1631
2661 !
946
3112
2680
1171
1782
2432
1019
2881
2807
1114
1964
2202
1102
2650
2946
1057
2126
2029
1200
2422
3096
1000
2318
1859
1319
2193
3264
944
2464
1745
1424
2021
160°.
1025
3579
2629
1630
1548
1851
1093
3350
2821
1516
1642
1737
1175
3118
3042
1402
1748
1623
1264
2887
3301
128 8
1868
1510
1371
2656
3609
1175
2006
1396
1497
2427
3783
1117
2167
1283
1647
2197
3975
1061
2356
1170
1783
2025
4191
1003
2463
1113
1939
1855
4430
947
2579
1056
2061
1741
280°.
1314
3590
2705
1000
2198
1627
1404
3360 
100°.
892
3517
2357
1513
1507
3128 1
936
3345
2535
1399
1626
2895
1002
3113
2747
128 5
1765
2664
1081
2882
2996
1172
1928
2435
1171
2652
3140
1115
2124
2204
1275
242 3
3295
1059
2301
2031
1402
2194
3468
1001
2509
I860
1516
2022
3664
945
2670
1746 !
1647
1852
200°.
1121
3583
2850
1632 !
1749
1738
1195
3353
3056
1518
1862
1624
1283
3121
3296
1404
1993
1511
1382
2890
3578
1289
2141
1397
1501
2659
3917
1176
2316
1284
1639
243
4104
1119
2519
1170
1805
219 9
4321
1062
2638
1113
1953
2027
4555
1004
2763
1057
2128
1857
4815
948
2902
1000
2264
1743
3058
943 
.
Thermal Properties of Normal Pentane. 359
Relation of Pressure to Temperature at Constant Volume.
Isochors.
For the smaller volumes isobars were first constructed
from the isothermals, and the temperatures at definite volumes
were read from the isobars. The data from which the
isobars were constructed are given below : —
Isobars read from Isothermals.
Temp....
<
' 130°.
140°.
150°.
160°.
170°.
180°.
190°.
200°.
210°.
220°.
230°.
Pressure
in
Volume in cub. cms.
metres.
12
20066
20796
21692
16
19958
20639
21464
22546
20
19852
20500
21266
22240
23526
25600
24
19755
20372
21094
21986
23115
24715
27700
28
19661
2 0254
20935
21767
22780
24105
26145
30840
32
19574
20144
2 0790
21574
22496
23650
25255
27915
36
T9486
20042
20656
21398
22247
23285
24635
26602
29910
40
19406
19948
20532
21256
22026
22965
24160
25775
28100
31960
44
19330
19861
20417
21087
21830
22695
23772
25135
27000
29660
48
19257
19776
20307
20946
21652
22460
23438
24630
2' 61 80
28290
31180
52
19186
20204
, 20816
21487
22245
23150
24225
25575
2 7330
; 2 9590
56
20106
20691
21335
...
26570
 28460
111 the following tables the data for the isochors are given ;
those for small volumes were read from the isobars, and those
for larger volumes from the isotherms.
Isochors read from Isobars.
Volume.
20.
21.
22,
23.
24.
25.
26.
27.
28.
29.
30.
Pressure
in
Temperature.
metres.
12
1290
1425
16
1307
1447
15535
16345
1696
20
1324
1468
1578
1663
17285
17775
24
13415
1488
1601
1691
1760
1813
18535
1884
1906
1921
1931
28
1360
1508
1625
17185
17925
18505
18945
19255
1950
19725
1989
32
1377
1529
1650
17465
1824
18865
1935
1974
20015
36
1394
1549
1672
1775
18575
19205
1974
2015
20495
2078
2102
40
1410
1568
1698
1804
18885
1957
20125
20585
2097
21275
2156
44
1429
1588
17205
18305
1918
199 05
2051
2100
2143
2179
2209
48
14455
16095
17465
1859
1950
2026
2089
21425
2188
22275
2263
52
1463
16295
1768
1883
1980
2060
21275
2183
2232
2277
2315
56
...
I
16495
2224
22775
2324
...
360
Mr. J. Rose limes and Dr. 8. Young on the
Isochors
read from
Isothermals
Volume.
29.
30.
32.
34.
36.
38. i 40.
1
43. 46.
1
50.
Temp.
Pressure.
190
22500

200
29920
28720
27330
26760
26500
26360
26290
26200
26150
26090
210
37770
35950
33600
32260
31480
30960
30580
30140
29800
29410
i 220
45800
43210
39960
37930
36600
35660
34960
34120
33450
32720
230
53940
50810
46430
43730
41820
40420
39400
38150
37140
36000
240
53040
49870
47390
45450
44000
42250
40890
39370
250
52830
50470
48630
46400
44630
42720
260
53240
50550
48290
45940
270
54670
52050
49260
280
j
1
1
55800
52500 ,
Isochors read from
Isothermals
Volume.
5  5.
6. I 65.
1
7.
8.
»
10. ! 12.
1
14.
16.
Temp.
Pressure.
180
1 "
18600
17335
16130
190
•■
200
25980
25760
25450
25140
24400
23540
22630
20840
19150
17690
210
i 28940
28460
220
! 31920
31090
30350
29630
28120
26730
25400
23000
20985
19200
230
1 34800
33750
240
! 37790
36400
35140
33940
3 750
29830
281 ib
2 150
22730
20680
250
! 40740
39000
260
! 43590
41510
39770
38140
3 300
32870
30790
27260
24460
22140
270
1 46400
44040
280
49330
46600
44270
42220
38740
35860
33390
29270
26110
23555
Isochors read from Isothermals.
Volume.
18. j 20.
22.
26.
30.
35.
40.
50.
60.
70.
80.
Temp.
Pressure.
100
3965
3532
120
.. ...
6725
5640
4849
4249
3782
140
9880
8985
8025
7250
6055
5190
4535
4026
160
13640
12865
12115
10825
9735
8635
7772
6465
5515
4812
4263
180
15030
14060
13180
11695
10475
200
16370
15210
14215
12520
11180
9830
8775
7*250
6150
5345
4726
220
17695
16370
15220
13330
11880
240
18980
17505
16250
14160
12570
11020
9800
8010
6805
5900
5208
260
20225
18610
17230
14980
13260
280
21465 19690
18200
15790
13975
12200
10825
8815
7462
6465
5699
Thermal Properties of Normal Pentane.
Isochors read from Isothermals.
361
Volume.
90.
100.
120.
140.
160.
180.
200.
230.
260.
300.
350.
Temp.
Pressure.
o .
1 40
857
i 60
1485
1347
1182
1053
920
80
2705
2304
2001
1773
1589
1440
1260
1123
981
100
3187
2907
2464
2137
1889
1693
1532
1340
1193
1040
896
120
3404
3098
2620
2272
2005
1794
1622
1419
1262
1100
947
140
3613
3285
160
3832
3477
2931
2534
2233
1996
1804
1576
1400
1219
1048
200
4237
3832
3231
2791
2458
2196
1981
1729
1535
1334
1146
240
4653
4212
3536
3047
2679
2393
2i57
1882
1669
1451
1246
280
5087
4588
3836
3305
2905
2592
2337
2038
1808
1571
1349
The values of b and a in the equation p = bT — a were
obtained graphically from the preceding data. As with iso
pentane, the deviations are exceedingly small at the largest
and smallest volumes and about the critical volume, but are
larger at intermediate volumes ; they exhibit a similar regu
larity and are in the same direction as with isopentane. Here
again the relation p = bT — a at constant volume, if not abso
lutely true, may be taken as a very close approximation to the
truth.
In studying the variation of b and a with the volume it
w r as found convenient, in the case of isopentane, to plot the
. , 10,000 . e 10 10 . , i , ... , . .
values or — ] and or — T against v 3 : and this has also been
V ^r, _! 10*
done for normal pentane. The values of b } a, v 3 ; y— , and
10 10 .
for a series of volumes are given in the table below and,
10*
av
for the sake of comparison, the corresponding values of
10 io bv
—  for isopentane are added
values of 10 10 /av 2
(Table p. 362.)
plotted against v
in the
and
av"
The values ol .
diagram on p. 363.
In a former paper by one of the authors (Phil. Mao, xliv.
p. 77) it was pointed out that, besides the quantities b and a,
it is often useful to consider a fresh quantity r, which is
defined as follows : — For each volume there is one and onlv
one temperature at which the gas has its pressure equal
to that given by the laws of a perfect gas : this temperature
is denoted by t. It is also shown that the numerical value
362 Mr. J. Roselnnes and Dr. S. Young on the
b.
a.
10
jbv.
10 10
jav 9 :
Vol.
in c.c.'s.
i
20
.From drawn isockors.
N.
Pentane.
Iso
pentane.
N.
Pentane.
Iso
pentane.
2312
917,550
•7937
2163
2165
2725
2783
21
1980
811,210 
•7809
2405
2487
2795
2952
22
1608
698,520
•7689
2725
2764
2958
3053
23
1436
610,860
•7576
3 028
3051
3095
3181
24
1265
544,000
•7468
3294
3365
3191
3329
25
1132
490,380
•7368
3534
3602
3263
3397
26
1010
438,880
•7272
3808
3851
3371
3480
27
940
409,900
•7181
3940
4056
3347
3522
28
858
373,910
•7095
4162
4279
3411
3586
29
790
343,580
•7013
4365
4520
3461
3670
30
730
316,470
•6931
4566
4731
3511
3728
32
6426
276,720
•6786
4863
4986
3529
3708
34
5727
244,270
•6650
5136
5281
3541
3736
36
5238
221,500
•6526
5303
5455
3484
3682
38
4786
200,200
•6408
5499
5507
3459
3632
40
4486
186,100
•6300
5573
5733
3358
3553
43
4073
166,640
•6150
5710
5859
3245
3426
46
3712
149,540
•6013
5856
5973
3160
3313
50
3311
130,570
•5848
6 040
6114
3009
3134
55
2927
112,470
•5665
6212
6298
2939
3062
G0
2604
97,320
•5503
6400
6489
2855
2976
65
2347
85,350
•5358
6555
6628
2773
2881
7
2126
75,210
•5227
6720
6786
2713
2816 ;
8
179'5
60,377
•5000
6964
7046
2588
2697
9
1539
49,178
•4805
7220
7294
2510
2615
10
1345
40,923
•4642
7435
7541
2444
2564
12
1075
30,043
•4368
7752
7917
2312
2466
14
8835
22,639
•4149
8085
8258
2254
2405
16
7415
17,388
•3968
8429
8463
2247
2327
18
6515
14,478
•3816
8524
8698
2132
2288
20
5690
11,721
•3684
8787
8872
2133
2242
22
5100
9,938
•3569
8913
8979
2079
2172
26
4150
7,125
•3375
9268
9314
2076
2160
30
3535
5,570
•3218
9430
9470
1995
2079
35
2978
4,261
•3057
9594
1916
40
2547
3.263
•2924
9815
9813
1915
1969
50
1973
2,089
•2714
1014
1026
1915
2067
60
1623
1,521
•2554
1027
1046
1826
2046
70
1375
1,143
•2426
1039
1053
1785
1957 i
80
1191
894
•2321
1050
1049
1748
1788
90
1045
694
•2231
1063
1061
1779
1782
100
9325
563
•2154
1072
1074
1775
1845
120
7653
388
•2027
1089
1087
1790
1823
140
6520
296
•1926
1096
1092
1720
1747
160
5677
230
•1842
1101
1107
1700
1860
180
5 030
187
•1771
1104
1118
1650
1930
200
4478
138
•1710
1117
1124
1810
1950
230
3892
113
•1632
1115
1118
1670
1730
260
3418
815
•1567
1125
1122
1810
1700
300
2953
620
•1496
1129
1127
1790
1680
350
2513
40
•1419
1137
11*33
2040
1770 j
Thermal Properties of Normal Pentane.
363
G O
D ^„
c (
&
O
p0°
G °
O
3
3
o<
08 c
<
o
o
r
€
3
O
1500
2000
25C0
3000
lOio
of t is given by the expression
; making use of the
bR/v
values of a and b already given for normal pentane, the values
of t have been calculated, and the results are given in the
folio win £ Table : —
V.
r.
v.
T.
v.
35
T.
20
4880
46
8149
833 9
21
5171
50
8243
40
8410
22
5476
55
8288
50
848
23
5760
60
8354
60
8266
24
6010
65
8384
70
8106
25
6234
7
8432
80
7982
26
6474
8
8433
90
8136
27
6609
9
848 9
100
8171
28
6803
10
8501
120
8453
29
6980
12
8449
140
8433
30
7158
14
8492
160
8214
32
7423
16
8616
180
8060
34
7665
18
8432
200
8625
36
7802
20
8543
230
8248
38
7967
22
845 8
260
8402
40
7997
26
8595
300
8267
43
8070
30
8491
350
8696
364 Mr. J. KoseInnes and Dr. S. Young on the
An examination of this table shows that r remains fairly
constant for all large volumes down to about vol. 8. The
actual numbers obtained vary a good deal ; but these variations
are sometimes in one direction and sometimes in another, and
there is no steady increase or decrease. It appears, then,
that all the values of r above vol. 8 could be treated as the
same without introducing any serious error ; this occurred
likewise in the case of isopentane. What is still more note
worthy is that the same constant value of r could be used for
both normal pentane and isopentane, keeping within the
limits of experimental error. The mean value of t for all
volumes above 8 was found to be 842*4 for isopentane ; it is
838*5 for normal pentane ; and the intermediate value 840
could be used in both cases without introducing any error
greater than the unavoidable errors of experiment.
When we pass on to the neighbourhood of the critical
point, the value of r diminishes steadily as the volume
decreases. For the critical volume itself t is about 807, and
for vol. 2 it has sunk to 488.
The most important conclusion arrived at in the case of
isopentane was that the molecular pressure a does not follow
a continuous law, but passes abruptly from one law to
another somewhere about vol. 3*4 (Phil. Mag. xliv. p. 79).
This inference was based on the study of a diagram in which
the quantity — ^ was plotted against t>~*, and there appeared
to be considerable evidence of discontinuity in the neighbour
hood of the volume already mentioned. Of course it is
impossible to prove discontinuity of slope by means of a
series of isolated points, but it is suggested very strongly ;
and even if there be not discontinuity in the true mathematical
sense of the term, there seems to be such a rapid change of
behaviour as to amount practically to the same thing.
It was therefore a matter of some interest to discover
whether the diagram obtained by plottino —  against v~i in
the case of normal pentane would exhibit the same peculiarity.
The diagram is given on p. 363, and it is easily seen that we
have here a similar suggestion of discontinuity in the slope of
— 2* this occurs somewhere about vol. 3*4, as with isopentane.
In attempting to find a formula for the pressure of normal
pentane we are therefore confronted with the possibility that
we may require two distinct algebraic equations. We may
simplify the problem considerably by confining our attention
Thermal Properties of Normal Pentane.
365
to volumes lying above 3*4 ; and this limitation still leaves us
with all those conditions of the substance in which we can
most usefully compare it with isopentane.
Looking at the table on p. 362, which gives the series of
values of — # an( * comparing it with the similar table for
isopentane (Proc. Phys. Soc. xiii. pp. 654, 655), we notice
that at the same volume the value of — t, is always smaller in
the former case than in the latter. The difference is not great,
but it remains too persistently with the same sign for us to
disregard it. As we proceed to larger and larger volumes,
however, the difference diminishes on the whole, and an
interesting question arises whether we should be justified in
treating it as ultimately vanishing when v is made infinite.
Sffi—
zoo
•>
©
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o ©
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o
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o
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2O0
p
I
To elucidate this point a diagram was drawn in which the
differences of — ^ between isopentane and normal pentane
were plotted against v~* ; this diagram is reproduced
above. The diminution in the differences with increase of
volume is well shown in spite of the " wobbling " at large
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 G
366 Mr. J. RoseTnnes and Dr. S. Young on the
o
volumes; and a line running through the points might
apparently end at zero difference. But though this result
might be accepted as consistent with the experimental
evidence if there were independent grounds for believing in
it, it cannot be considered as the most probable judging
solely from the diagram ; we should be rather led to believe
that even at infinite volumes the value of — : . for isopentane
remained larger than that for normal pentane.
The above results respecting a and t are chiefly interesting
because they seem capable of throwing some light on the
vexed question of the influence exerted by difference of
chemical structure on the thermal properties of a substance.
Concerning this matter very little is known at present ; but
it is common knowledge among organic chemists that two
substances may have the same chemical composition and
show practically the same behaviour whilst in the condition
of rare vapour, and yet they may differ considerably as to
their thermal properties in the liquid state. The great field
of observation in which the substances lie between the con
ditions of a rare vapour and a common liquid has been left
almost entirely unexplored. This gap in our knowledge
makes it impossible to say in what precise manner the differ
ence between two isomeric substances originates ; whether it
arises conjointly with the first deviations from Boyle's law,
or whether the difference remains inappreciable even with
increasing density until we reach the neighbourhood of the
critical point. We may put the problem more precisely as
follows : — If we imagine the pressure given by a series of
ascending powers of the density, what is the lowest power of
the density which has different coefficients for two isomeric
substances ?
We are now able to answer this question with a fair amount
of exactness in the case of the two isomers, normal pentane and
isopentane. If, as seems most probable, there is a difference
between the — ^ for normal pentane and the — T for iso
pentane, even at infinitely large volumes, this shows that the
coefficients of the second power of the density in the expan
sion of p must be different for the two substances. On the
other hand, if there is no difference between the — « for
j avr
normal pentane and the — ^ for isopentane at infinitely large
volumes, then the coefficients of the second power of the
density in the expansion for p must be the same, since t has
Thermal Properties of Normal Pentane. 367
already been shown to have the same value for the two sub
stances at infinite volumes ; and the lowest power of the
density which has different coefficients for the two isomers
must be the third.
It w r as thought advisable to test these conclusions by a
different method. In a former paper by one of the authors
(Phil. Mag. xliv. p. 80 ; see also Phil. Mag. xlv. p. 105) it
was shown that in the case of isopentane we might reproduce
the original observations very closely by putting
* v \ v + k — gv~ 2 )
gv~ 2 ) v(v + k) 9
where R, <?, k, g, and I are constants. If we assume that this
formula holds also for normal pentane, and if it be true that
the difference of pressure between normal pentane and iso
pentane at the same temperature and volume varies ultimately
as the third powder of the density, then we should be able to
reproduce the experimental data for normal pentane by
means of the above formula, giving to R, e, and I the values
already found for isopentane. We may accordingly take
11 = 86356, £ = 7473, 1= 5420800, and we still have the
constants k srndg left at our disposal to meet the requirements
of the normal pentane data. On examining the observations
given in Series I. of this paper we find that we can con
veniently put k — 3*135, #=6'695, and we have to test how
far the formula with these constants reproduces the experi
mental results given in Series II., III., and IV. In order to
institute an effective comparison between theory and observa
tion a diagram was made in which pv was plotted against v~i ;
the calculated isothermals were drawn as continuous lines,
while the experimental values were put in as dots. An
examination of the diagram shows that a fair concordance
between calculation and experiment has been secured ; but
the agreement is not so good as could be wished. Deviations
amounting to 1 per cent, are not uncommon, and in places
they approach 2 per cent. If we have regard to the differ
ences which often occur in inquiries of this kind between the
results of independent observers, we might conclude that the
above deviations are unimportant, and that R, e, and I were
really the same for the two pentanes as supposed. But it
seems more likely that the deviations are too large to be
neglected ; hence, the most probable inference is that the I
for normal pentane is not the same as the I for isopentane,
thereby confirming our former conclusion.
2 C2
[ 368 ]
XXXII. An Application of the Diffraction Grating to Colour
Photography. By B. W. Wood*.
IF a diffractiongrating of moderate dispersion and a lens
be placed in the path of a beam of light coming from a
linear source, and the eye be placed in any one of the spectra
formed to the right and left of the central image, the entire
surface of the grating will appear illuminated with light of a
colour depending on the part of the spectrum in which the
eye is placed. If one part of the grating has a different
spacing from the rest, the spectrum formed by this part will
be displaced relatively to the first; and if the eye be placed
in the overlapping part of the two spectra, the corresponding
portions of the grating will appear illuminated in different
colours. This principle I have made use of in the develop
ment of a new method for producing photographs in natural
colour. I have eliminated the use of pigments and coloured
screens entirely in the finished picture, the photograph being
nothing more nor less than a diffractiongrating of variable
spacing, the width between the lines in the different parts of
the picture being such as to cause them to appear illuminated
in their proper colours when view r ed in the manner described.
We will take at the start three diffractiongratings of such
spacing that the deviation of the red of the first is the same
as that of the green of the second and the blue of the third
(the red, green, and blue in question being of the tints of the
primary colours of the YoungHelmholtz theory of colour
vision). If these three gratings be mounted side by side in
front of a lens their spectra will overlap; and an eye placed
in the proper position will see the first grating red, the second
green, and the third blue. If the first and second be made
to overlap, this portion will send both red and green light to
the eye, and will in consequence appear yellow. If all three
be made to overlap in any place, this place will send red,
green, and blue light to the eye, and will appear white.
The method that I first employed to produce photographs
showing natural colours on this principle is the following 5—
Three negatives were taken through red, green, and Mue
screens in the usual manner : from these, positives were made
on ordinary lanternslides (albumenslides are necessary for
reasons which I will speak of presently) . The positives, when
dry, were flowed with bichromated gelatine and dried in
subdued light. The three diffractiongratings of proper
spacing, ruled or photographed on glass, w^ere placed over
these positives, and exposed to the sun or electric light for
* Communicated bv the Author.
Diffraction Grating and ColourPhotography . 369
thirty seconds. On washing these plates in warm water,
diffractiongratings of great brilliancy were formed directly
on the surface of the film. Albumenplates must be used,
since the warm water softens and dissolves a gelatine film.
Three sheets of thin glass, sensitized with the bichromated
gelatine, were placed under the three positives, and prints
taken from them. The portions of each plate on which the
light had acted bore the impression of the corresponding
diffractiongrating, strongly or feebly impressed, according to
the density of the different parts of the positives. These three
plates, when superimposed and placed in front of a lens and
illuminated by a narrow source of light, appear as a correctly
coloured picture, when viewed with the eye placed in the
proper position. Perfect registration of the different parts
of the picture could not be obtained in this way, for obvious
reasons. I worked for awhile with the thin glass from which
covers for microscopical slides are made. This gave much
better results, but was too fragile for practical purposes. It
then occurred to me that if 1 could get the entire grating
system on a single film, not only would the difficulty about
perfect registration vanish, but the pictures could be repro
duced by simple contactprinting on chromgelatine plates as
easily as blue prints are made. I was surprised to find that
successive exposures of the same plate under the positives,
perfect registration being secured by marks on the plates,
produced the desired result. On washing this plate in warm
water and drying, it becomes the finished coloured photograph.
Where the reds occur in the original, the spacing of the first
grating is present; where the yellows occur the spacings of
both the first and second are to be found superimposed ; where
the blues occur are the lines of the third grating ; while in
the white parts of the picture all three spacings are present.
It seems almost incredible that, by exposing the plate in suc
cession under two gratings the spacings, of both should be
impressed — superimposed — in such a manner as to give the
colours of each in equal intensity ; but such is the feet. Thus
far I have had at my disposal but two gratings of only ap
proximately the right spacing, one giving the red, the other
the green : with these I have photographed stained glass
windows, birds, and butterflies, and other stilllife objects,
the finished pictures showing reds, yellows, and greens in a
most beautiful manner. By making a separate plate from
the blue positive, using the same spacing as with the green,
and setting this plate behind the other at an angle, 1 have
obtained the blues and whites, the gratingspace being dimin
ished by foreshortening, though, of course, perfect regis
370 Mr. R. W. Wood on an Application of
tration of the different portions of the picture could not be
obtained.
One of the great advantages of this method is the facility
with which duplicates can be made. If we place the finished
picture in a printingframe over a glass plate coated with
bichromated gelatine, and expose it to sunlight, on washing
the plate in warm water we obtain, by a single printing
process, a second colourphotograph, equal to the first in every
respect, and also positive. From this second copy we can
print others, all being positive.
The apparatus for viewing the pictures consists of a cheap
doubleconvex lens mounted on a little frame, as shown in
fig. 1, with a perforated screen for bringing the eye into the
right position. I find that, by using a lens of proper focus,
it is possible to so adjust the apparatus that the picture can be
seen in its natural colours with both eyes simultaneously,
since corresponding overlapping spectra are formed on each
side of the central direct image. A gas flame turned edge
wise, or the filament of an incandescent light, makes a con
venient source of light. The colours are of great brilliancy
and purity, almost too brilliant in fact, though dark reds and
ochres are reproduced with considerable fidelity. The pictures
can be projected by employing a powerful arclight, placing
a rather wide slit in the overlapping spectra, and mounting
the projecting lens beyond this. The pictures that I have
obtained thus far measure 2'5 in. by 2*5 in., and have been
thrown up about 3 feet square. The fact that only a small
percentage of the light is utilized makes great amplification
difficult. Certain experiments that I have made lead me to
believe that the process can be greatly simplified.
I have exposed an ordinary photographic plate in a camera
the Diffraction Grating to Colour Photography, 371
under a diffractiongrating placed in front of, and in contact
with, the film. On development, we obtain a negative the
dark portions of which are broken up into fine lines, corre
sponding to the lines of the grating; and on viewing this in
the apparatus just described, the blue components of the picture
are seen, though not so brilliant as with the transparent
gelatine plate owing to the coarseness of the grain.
I believe that by the use of a suitable photographic plate to
be exposed in succession in the camera, under red, green, and
blue screens, on the surfaces of which suitable diffraction
gratings have been photographed, it will be possible to obtain
the colourphotograph directly. The screens can be swuno
into position in succession by a suitable mechanical arrange
ment operated outside of the camera. The plate, on deve
lopment, will be a negative in the ordinary sense of the term,
though when seen in the viewingapparatus it will appear as
a coloured positive, since on the transparent portions which
correspond to black in the original, no grating lines have
been impressed : consequently these portions will appear dark.
The dark portions, however, where the lines are impressed will
light up in their appropriate colours. From this plate as
many copies as are desired can be made by contactprintino
on bichromated gelatine.
Of course it is a question whether superimposed gratinos
can be impressed on a plate in this manner. Judging from
the experiments I have made, I imagine that the gratings on
the colourscreens would have to be made with the opaque
portions broad in proportion to the transparent.
I have overcome the difficulty of obtaining large diffraction
gratings by building up photographic copies in the followino
manner. The original grating ruled on glass was mounted
against a rectangular aperture in a vertical screen, the lines
of the grating being horizontal. Immediately below this was
placed a long piece of heavy plateglass, supported on a slab of
slate to avoid possible flexure. A strip of glass, a little wider
than the grating, sensitized with bichromated gelatine was
placed in contact with the lines of the grating, and held in
position by a brass spring. The lower edge of the strip rested
upon the glass plate so that it could be advanced parallel to
the lines of the grating, and successive impressions taken by
means of light coming through the rectangular aperture. In
this way I secured a long narrow grating; and by mounting
this against a vertical rectangular aperture, and advancing a
second sensitized plate across it in precisely the same manner,
1 obtained a square grating of twentyfive times the area of
the original. It was in this manner that I prepared the
e72 Dr. G. Johnstone Stoney on
orating used to print the impressions on the three positives.
So well did they perform, that it seemed as if it might be
possible in this way to build up satisfactory gratings of large
size for spectroscopic work. Starting with a 1inch grating
of 2000 lines, I have bailt up a grating 8 inches square,
which, when placed over the objectglass of a telescope,
showed the dark bands in the spectrum of Sirius with great
distinctness. No especial precautions, other than the use of
the flat glass plate, were taken to insure absolute parallelism
of the lines, and I have not Lid time to thoroughly test the
grating. The spectra, however, are of extraordinary brilliancy;
and on the whole the field seems promising. This matter will,
however, be deferred to a subsequent paper.
Physical Laboratory of the
University of Wisconsin, Madison.
XXXIII. Denudation and Deposition.
By G. Johnstone Stoney, M.A., D.Sc, F.R.S*
IN a lecture to the Royal Geographical Society, of which a
copious extract is given in ' Nature ' of the 2nd of
February, 1899, Dr. J. W. Gregory discusses many of the
causes which may have led to the existing form of the earth.
But there is one important factor in the problem left unnoticed,
namely, the conspicuous alterations of level which may be
attributed to the earth's compressibility, and which seem to
have been brought about wherever either denudation or
deposition have continued over wide areas and for a long time.
Dr. Gregory makes a convenient division of the earth into
three parts : — (1) the unknown internal centrosphere ; (2) the
rocky crust or litho sphere ; (3) the oceanic layer, or hydro
sphere. These, with the atmosphere, which may be added as
a fourth part, make up the whole earth.
If now we imagine a pyramid whose base is a square centi
metre of the surface of the solid part of the earth and whose
vertex is the earth's centre, it has a volume of about 212
cubic metres, which is the same as 212 millions of cubic centi
metres. This pyramid passes first through the lithospheric
yhell, or outer crust, and then halfway across the centrosphere
to the centre of the earth. All the materials of which it
consists are compressible. Those which lie within the outer
shell consist mainly of carbonates, silicates, and aluminates,
a id have probably a coefficient of compressibility about equal
to that of glass ; while the compressibility of the centrosphere
is unknown, and may be either more or less. The observed
form of the earth's surface seems to suggest that the average
* Communicated by the Author.
Denudation and Deposition. 373
compressibility of the lithosphere and centrosphere taken
together is not far from that of the more incompressible kinds
of glass. Glass of this description yields to compression
about 2^ times more than solid cast iron, but less than
mercury (which seems to be the only liquid metal that has
been experimented on) in the ratio of 2 to 3. It is about
20 times more incompressible than water.
We shall then, as a provisional hypothesis, assume that the
earth has the same compressibility as the more resistant kinds
of glass, which lose about 2J billionths of their volume for
each pressure of a dyne per square centimetre over their
surface. Combining this with the volume of the earth
pyramid given above, we find that our hypothesis leads to the
conclusion that if the sides of the pyramid were kept from
yielding, and if the weight of a cubic centimetre of water
were placed on its outer end, this would reduce its hulk by
half a cubic centimetre. A cubic centimetre of stone, of
specific gravity 3, would accordingly depress its outer end by
1J centimetres. It follows from this that if meteors rained
upon the earth (supposed to be without an ocean) producing
a deposit over its whole surface a centimetre thick, and of
material as dense as stone., the result would be that the earth
after this accession would be smaller ; its surface would sink
down about half a centimetre. Correspondingly, if by any
agency a centimetre of the earth's crust could be removed over
the whole earth, the earth's surface would stand J a centimetre
higher than before. These are the effects which deposition and
denudation would respectively produce if they could operate
over the whole earth. And, if they operate over any extensive
area of the earth's surface ; they will produce effects of the same
kind, complicated a little by the displacement of the earth's
centre of attraction, or rather locus of centres of attraction.
This may be well seen in the oldest parts of the oldest
continents — parts of Asia and Africa — to whose present
elevation denudation *, operating over an extensive area and
for long ages, has probably chiefly contributed. And, corre
spondingly, there is a deepening of those parts of the ocean
where the deposition of sufficiently heavyt material has been
going on over a great area for an immense time.
* Underground waters produce the same dynamical eifect as surface
denudation, by reason of the materials they remove in solution.
f Yv^here the subaqueous deposit is spread over only a small part of
the surface of the globe (which is the only case we need consider), the
compression is due, not to the whole weight of the deposit, but only to
its excess over the weight of an equal bulk of water. Hence to produce
an equivalent effect the material must be denser than it would need to
be if the deposition had been on land.
374 On Denudation and Deposition.
The extent of the area is an essential condition, L e., the
lateral dimensions of the inverted pyramid which has the area
for its base and the centre of the earth for its vertex. If the
area is small or narrow, oblique forces exerted by the parts
surrounding this pyramid come more into play. They enable
the part within the pyramid to act like a bridge ; and the
support thus given enables denudation, if limited to a small
area, to scoop out valleys, and deposition to produce ridges,
as may be seen in the glaciers and moraines of mountainous
countries. On the other hand, if the erosion due to glacial
action takes effect over a great stretch of country, as it does
in Greenland, and as it formerly did in Ireland, it causes the
surface to rise.
A nearly even balance between the two opposite tendencies
may be seen in Egypt, where borings exhibit fluviatile de
posits at great depths below the present surface, although the
surface is only about as much raised above the sea now as it
was when those ancient deposits were laid down by the Nile.
Each year's deposit makes the surface go down, but only
about as much as its own thickness, so that the new surface
each year is not far from being at the same level as that of
the preceding year. If the deposit had taken place over a
much greater breadth of country, the whole would have gone
down. It would have become a ridge if it had been confined
to a much narrower strip and if the river could have been
kept from diverging.
A similarly instructive case is that of Brazil, where an
immense plateau is continuously being denuded by the vast
rivers that drain it. But here there is also an equally un
interrupted addition to the solid materials of the earth by the
luxuriant tropical vegetation which everywhere prevails ; and
it is probably because the accessions and withdrawals are
nearly equal to one another, that the level of the surface has
been but little changed.
Denudation may cause the surface to rise within a space
which is in a considerable degree more circumscribed than
the areas of elevation hitherto considered, if the conditions
are such that the stresses that come into existence round the
boundary of this limited space can produce faults, and pre
vent the material which is outside the pyramid from being in
a position to help to keep down the material which is within.
This seems to have happened in the case of that vast mass of
mountains — the Himalayas, the Hindu Kush, and their asso
ciated ranges — where excessive denudation accompanied by
the isolation secured by faults has occasioned a proportionately
great elevation above what was probably a humble beginning ;
On Transmission of Light through an Atmosphere. old
where the deposits in the Bay of Bengal are probably the
cause of its great depth; and where earthquakes in the
intervening regions betray when the faults are establishing
themselves which render the rising and the descending areas
independent of one another, and allow the denudation on the
one side and the deposition on the other to produce each its
full effect, without mutual interference.
Of course all compressions and dilatations must be accom
panied by other movements Avithin the earth, and at all
depths ; which may be slow but are no less sure. In fact,
there is no material which can resist yielding to differences
of pressure, however feeble, if they act for a long time and
over a large surface ; and such pressures, urging in various
directions, must arise both from the compressions and dilata
tions spoken of above, and from other causes, among which
movements of heat and the heterogeneous character of the
materials of which the earth consists are prominent. The
earth, therefore, is in a state of neverending change, which
to become conspicuous to man would only need to be placed
in some kind of kinematograph arrangement which would
hurry over millions of years in fractions of a second. These
effects mix with and complicate those which have been taken
account of in the present paper.
It is interesting to note how the agencies we have been
considering would operate upon other bodies of the universe.
Events equivalent to denudation and deposition which cause
excessively slow movements in our small earth, would act w T ith
increased promptness upon such great planets as Jupiter,
Saturn, Uranus, and Neptune, and with violence upon bodies
that attain the size of the sun and stars. On the other
hand, on bodies with the dimensions of the moon they are
relatively feeble, and must be very slow in producing any
appreciable effect.
XXXIV. On the Transmission of Light through an Atmosphere
containing Small Particles in Suspension, and on the Origin
of the Blue of the Sky. By Lord Rayleigh, F.R.S*
rpHIS subject has been treated in papers published many
J years ago . I resume it in order to examine more
closely than hitherto the attenuation undergone by the
primary light on its passage through a medium containing
small particles, as dependent upon the number and size of the
particles. Closely connected with this is the interesting
* Communicated by the Author.
t Phil. Mag. xli. pp. 107, 274, 447 (1871) ; xii. p. 81 (1881).
376 Lord Rayleigh on the Transmission of Light through
question whether the light from the sky can be explained by
diffraction from the molecules of air themselves, or whether it
is necessary to appeal to suspended particles composed of
foreign matter, solid or liquid. It will appear, I think, that
even in the absence of foreign particles we should still have a
blue sky *.
The calculations of the present paper are not needed in
order to explain the general character of the effects produced.
In the earliest of those above referred to I illustrated by
curves the gradual reddening of the transmitted light by
* My attention was specially directed to tins question a long while ao'o
by Maxwell in a letter which I may be pardoned for reproducing here.
Under date Aug. 28, 1873, he wrote :—
" I have left your papers on the light of the sky, &c. at Cambridge,
and it would take me, even if I had them, some time to get them assimi
lated sufficiently to answer the following question, which I think will
involve less expense to the energy of the race if you stick the data into
your formula and send me the result
" Suppose that there are N spheres of density p and diameter s in unit
of volume of the medium. Find the index of refraction of the compound
medium and the coefficient of extinction of light passing through it.
"The object of the enquiry is, of course, to obtain data about the size
of the molecules of air. Perhaps it may lead also to data involving the
density of the aether. The following quantities are known, being com
binations of the three unknowns,
M = mass of molecule of hydrogen ;
N= number of molecules of any gas in a cubic centimetre at G° C.
and 760 B.
s = diameter of molecule in any gas: —
Known Combinations.
MN = density.
Ms 2 from diffusion or viscosity.
Conjectural Combination.
—  = density of molecule.
" If you can give us (i.) the quantity of light scattered in a given
direction by a stratum of a certain density and thickness ; (ii.) the
quantity cut out of the direct ray ; and (iii.) the effect of the molecules
on the index of refraction, which I think ought to come out easily, we
might get a little more information about these little bodies.
" You will see by ' Nature,' Aug. 14, 1873, that I make the diameter
of molecules about ^Vo °f a wavelength.
" The enquiry into scattering must begin by accounting for the great
observed transparency of air. I suppose we have no numerical data
about its absorption.
"But the index of refraction can be numerically determined, though
the observation is of a delicate kind, and a comparison of the result
with the dynamical theory may lead to some new information."
Subsequently he wrote, " Your letter of Nov. 17 quite accounts for the
observed transparency of any gas." So far as I remember, my argument
was of a general character onfy.
an Atmosphere containing Small Particles in Suspension. 377
which we see the sun a little before sunset. The same
reasoning proved, of course, that the spectrum of even a
vertical sun is modified by the atmosphere in the direction of
favouring the waves of greater length.
For such a purpose as the present it makes little difference
whether we speak in terms of the electromagnetic theory or
of the elastic solid theory of light ; but to facilitate compari
son with former papers on the light from the sky, it will be
convenient to follow the latter course. The small particle of
volume T is supposed to be small in all its dimensions in
comparison with the wavelength (X), and to be of optical
density D' differing from that (D) of the surrounding
medium. Then, if the incident vibration be taken as unity,
the expression for the vibration scattered from the particle in
a direction making an angle 9 with that of primary vibra
tion is
— ^— _ S in cos — (btr)*, . . . (1)
r being the distance from T of any point along the secondary ray.
In order to find the whole emission of energy from T we
have to integrate the square of (1) over the surface of a
sphere of radius r. The element of area being 2wr 2 sin 6d0,
we have
^ S ^2nr*sm6d0 = ±7rP\m*ddd= ~;
so that the energy emitted from T is represented by
8tt s (D'D) 2 T* f9 .
"3" L>* \^ {Z)
on such a scale that the energy of the primary wave is unity
per unit of wavefront area.
The above relates to a single particle. If there be n
similar particles per unit volume, the energy emitted from a
stratum of thickness dee and of unit area is found from (2) by
introduction of the factor ndx. Since there is no waste of
energy on the whole, this represents the loss of energy in the
primary wave. Accordingly, if E be the energy of the pri
mary wave,
lc/E_ 87r 3 n (D'D) g T a
e dx ~ a w \ 4 ; • • • ( °)
whence E = E 6 fe , (I)
where ; _ 8tt^ (D'D) 2 T^
8 D 9  ~\ 4 ' ' ' * ^
* The factor n was inadvertently omitted in the original memoir.
378 Lord Rayleigh on the Transmission of Light through
If we had a sufficiently complete expression for the scattered
light, we might investigate (5) somewhat more directly by
considering the resultant of the primary vibration and of
the secondary vibrations which travel in the same direction.
If, however, we apply this process to (1), we find that it fails
to lead us to (5), though it furnishes another result of interest.
The combination of the secondary waves which travel in the
direction in question have this peculiarity, that the phases
are no more distributed at random. The intensity of the
secondary light is no longer to be arrived at by addition of
individual intensities, but must be calculated with considera
tion of the particular phases involved. If we consider a
number of particles which all lie upon a primary ray, we see
that the phases of the secondary vibrations which issue along
this line are all the same.
The actual calculation follows a similar course to that by
which Huygens' conception of the resolution of a wave into
components corresponding to the various
parts of the wavefront is usually veri
fied. Consider the particles which oc
cupy a thin stratum dx perpendicular
to the primary ray x. Let AP (fig. 1)
be this stratum and the point where
the vibration is to be estimated. If
AF = p, the element of volume is
dx . "27rpdp, and the number of particles
to be found in it is deduced by intro
duction of the factor n. Moreover, if
Fisr 1.
OP =
AO =
S—„2
= x 2 \p 2 , and
pdp = rdr. The resultant at of all the
secondary vibrations which issue from
the stratum dx is by (1), with sin equal to unity
.dx
p» D'D ttT
277
cos —  (bt — r) 2irrdr }
or
7 D'DttT . 2tt,, ,
n ax . — y\ — sm — [bt — x)
U A A,
(6)
To this is to be added the expression for the primary wave
27T
itself, supposed to advance undisturbed, viz., cos — (bt — x),
and the resultant will then represent the whole actual dis
turbance at as modified by the particles in the stratum dx.
It appears, therefore, that to the order of approximation
afforded by (1) the effect of the particles in dx is to modify
the phase, but not the intensity, of the light which passes
an Atmosphere containing Small Particles in Suspension. 379
them. If this be represented by
cos ^ (fa a 8), (7)
A
8 is the retardation due to the particles, and we have
8 = wT^(D / D)/2D (8)
If fju be the refractive index of the medium as modified by
the particles, that of the original medium being taken as
unity, b s =({i — l)dw, and
At l = nT(D , D)/2D (9)
If /j! denote the refractive index of the material composing
the particles regarded as continuous, D' /D = fi' 2 , and
/.l=i«T0i»l), (10)
reducing to
ftI = «T(/*'l) (11)
in the case where yJ — 1 can be regarded as small.
It is only in the latter case that the formulas of the elastic
solid theory are applicable to light. In the electric theory,
to be preferred on every ground except that of easy intelli
gibility, the results are more complicated in that when (//— 1)
is not small, the scattered ray depends upon the shape and
not merely upon the volume of the small obstacle. In
the case of spheres we are to replace (D / — D)/D by
3(K / K)/(K' + 2K), where K, K' are the dielectric constants
proper to the medium and to the obstacle respectively*; so
that instead of (10)
1 3tiT fS*l
^ 1= TAl ^
On the same suppositions (5) is replaced by
On either theory
( u '2_\ \2 T2
3nX 4 ' { J
a formula giving the coefficient of transmission in terms of
the refraction, and of the number of particles per unit volume.
We have seen that when we attempt to find directly from
(1) the effect of the particles upon the transmitted primary
wave, we succeed only so far as regards the retardation. In
* Phil. Mag, xii. p. 98 (1881). For the corresponding theory in the
case of an ellipsoidal obstacle, see Phil. Mag. vol. xliv. p. 18 (1897).
380 Lord Rayleigh on the Transmission of Light through
order to determine the attenuation by this process it would be
necessary to supplement (1) by a term involving
but this is of higher order of smallness. We could, however,
reverse the process and determine the small term in question
a posteriori by means of the value of the attenuation obtained
indirectly from (1), at least as far as concerns the secondary
light emitted in the direction of the primary ray.
The theory of these effects may be illustrated by a com
pletely worked out case, such as that of a small rigid and
fixed spherical obstacle (radius c) upon which plane waves of
sound impinge *. It would take too much space to give full
details here, but a few indications may be of use to a reader
desirous of pursuing the matter further.
The expressions for the terms of orders and 1 in spherical
harmonics of the velocitypotential of the secondary disturbance
are given in equations (16), (17), § 334. With introduction
of approximate values of 7 and 7 b viz.
we get
[f »] + DM =  % 3 (! + t) cos *(*')
+ ^( 1 ~t) sin *( a *')> • • ( 15 )
in which c is the radius of the sphere, and k = 2ir/\. This
corresponds to the primary wave
1$] = cos k(at + w), (16)
and includes the most important terms from all sources in the
multipliers of cos k(at—r), sin k(at r). Along the course of
the primary ray (fi= — 1) it reduces to
k 2 c z lk b c 6
[^o] + [fi]=^ cos ^^)+ gg^ sin *(o*r) • ( 17 )
We have now to calculate by the method of FresnePs zones
the effect of a distribution of n spheres per unit volume.
We find, corresponding to (6). for the effect of a layer of
thickness dx,
1irndx{\ke % sin k{at + «) 3 7 ^c 6 cos h{at + x)}. . (18)
To ihis is to be added the expression (16) for the primary
wave. The coefficient of cos k\at\x) is thus altered by the
* ' Theory of Sound,' 2nd ed. § 334.
an Atmosphere containing Small Particles in Suspension. 381
particles in the layer dx from unity to (l—^k 4 c 6 7rndx), and
the coefficient of s'm k (at + x) from to \kc ?Jr nndx. Thus, if
E be the energy of the primary wave,
dE/E=%k 4 c 6 7rndx;
so that if, as in (4), E =E e~ hx ,
h = l7rnk 4 c 6 (19)
The same result may be obtained indirectly from the first
term of (15). For the whole energy emitted from one sphere
may be reckoned as
£j_W(l+/.)ty=^p > . ■ • (20)
unity representing the energy of the primary wave per unit
area of wavefront. From (20) we deduce the same value of
h as in (19).
The first term of (18) gives the refractivity of the medium.
If 8 be the retardation due to the spheres of the stratum dx,
sin k8=^kc d irndXj
or h = ±Trnc z dx (21)
Thus, if fi be the refractive index as modified by the spheres,
that of the original medium being unity,
^l=j7rnc 3 = ip, (22)
where p denotes the (small) ratio of the volume occupied by
the spheres to the whole volume, This result agrees with
equations formerly obtained for the refractivity of a medium
containing spherical obstacles disposed in cubic order*.
Let us now inquire what degree of transparency of air is
admitted by its molecular constitution, i. e., in the absence
of all foreign matter. We may take A=6xl0~ 5 centim.,
fi — 1 = *0003; whence from (14) we obtain as the distance x,
equal to 1/h, which light must travel in order to undergo
attenuation in the ratio e : 1 ,
<z>=4!xl0 13 xn (23)
The completion of the calculation requires the value of n.
Unfortunately this number — according to Avogadro's law
the same for all gases — can hardly be regarded as known.
Maxwell f estimates the number of molecules under standard
* Phil. Mag. vol. xxxiv. p. 499 (1892). Suppose m = oo , o = oo .
t " Molecules," Nature, viii. p. 440 (1873).
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 D
382 Lord Rayleigh on the Transmission of Light through
conditions as 19 X 10 18 per cub. centim. If we nse this value
of », we find
# = 8*3xl0 6 cm. = 83 kilometres,
as the distance through which light must pass through air
at atmospheric pressure before its intensity is reduced in the
ratio of 2*7 : 1.
Although Mount Everest appears fairly bright at 100 miles
distance as seen from the neighbourhood of Darjeeling, we can
not suppose that the atmosphere is ns transparent as is
implied in the above numbers ; and of course this is not to
be expected, since there is certainly suspended matter to be
reckoned with. Perhaps the best data for a comparison are
those afforded by the varying brightness of stars at various
altitudes. Bouguer and others estimate about *8 for the
transmission of light through the entire atmosphere from a
star in the zenith. This corresponds to 8' 3 kilometres of air
at standard pressure. At this rate the transmission through
83 kilometres would be ("8) 10 , or "11, instead of 1/e or *37.
It appears then that the nctual transmission through 83 kilo
metres is only about 3 times less than that calculated (with
the above value of n) from molecular diffraction without any
allowance for foreign matter at all. And we may conclude
that the light scattered from the molecules would suffice to
give us a blue sky, not so very greatly darker than that
actually enjoyed.
If n be regarded as altogether unknown, we may reverse
our argument, and we then arrive at the conclusion that n
cannot be greatly less than was estimated by Maxwell. A
lower limit for n, say 7 X 10 18 per cubic centimetre, is some
what sharply indicated. For a still smaller value, or rather
the increased individual efficacy which according to the
observed refraction would be its accompaniment, must lead to
a less degree of transparency than is actually found. When
we take into account the known presence of foreign matter,
we shall probably see no ground for any reduction of
Maxwell's number.
The results which we have obtained are based upon (14),
and are as true as the theories from which that equation was
derived. In the electromagnetic theory we have treated the
molecules as spherical continuous bodies differing from the
rest of the medium merely in the value of their dielectric
constant. If we abandon the restriction as to sphericity, the
results will be modified in a manner that cannot be precisely
defined until the shape is specified. On the whole, however, it
does not appear probable that this consideration would greatly
affect the calculation as to transparency, since the particles
an Atmosphere containing Small Partides in Suspension. 383
must be supposed to be oriented in all directions indifferently.
But the theoretical conclusion that the light diffracted in a
direction perpendicular to the primary rays should be com
pletely polarized may well be seriously disturbed. If the
view, suggested in the present paper, that a large part of the
light from the sky is diffracted from the molecules themselves,
be correct, the observed incomplete polarization at 90° from
the Sun may be partly due to the molecules behaving rather
as elongated bodies with indifferent orientation than as spheres
of homogeneous material.
Again, the suppositions upon which we have proceeded
give no account of dispersion. That the refraction of gases
increases as the wavelength diminishes is an observed fact ;
and it is probable that the relation between refraction and
transparency expressed in (14) holds good for each wave
length. If so, the falling off of transparency at the blue end
of the spectrum will be even more marked than according to
the inverse fourth power of the wavelength.
An interesting question arises as to whether (14) can be
applied to highly compressed gases and to liquids or solids.
Since approximately (/jl — 1) is proportional to n, so also is
h according to (14). We have no reason to suppose that
the purest water is any more transparent than (14) would
indicate ; but it is more than doubtful whether the calcula
tions are applicable to such a case, where the fundamental
supposition, that the phases are entirely at random, is violated.
When the volume occupied by the molecules is no longer
very small compared with the whole volume, the fact that
two molecules cannot occupy the same space detracts from
the random character of the distribution. And when, as in
liquids and solids, there is some approach to a regular spacing,
the scattered light must be much less than upon a theory of
random distribution .
Hitherto we have considered the case of obstacles small
compared to the wavelength. In conclusion it may not be
inappropriate to make a few remarks upon the opposite
extreme case and to consider briefly the obstruction presented,
for example, by a shower of rain, where tbe diameters of the
drops are large multiples of the wavelength of light.
The full solution of the problem presented by spherical
drops of water w r ould include the theory of the rainbow,
and if practicable at all would be a very complicated matter.
But so far as the direct light is concerned, it would seem to
make little difference whether we have to do with a spherical
refracting drop, or with an opaque disk of the same diameter.
2D2
384 On Transmission of Light through an Atmosphere.
Let us suppose then that a large number of small disks are
distributed at random over a plane parallel to a wavefront,
and let us consider their effect upon the direct light at a
great distance behind. The plane of the disks may be divided
into a system of Fresnel's zones, each of which will by
hypothesis include a large number of disks. If a be the area
of each disk, and v the number distributed per unit of area
of the plane, the efficiency of each zone is diminished in the
ratio 1: 1 — va, and, so far as the direct wave is concerned,
this is the only effect. The amplitude of the direct wave is
accordingly reduced in the ratio 1 : 1— va, or, if we denote the
relative opaque area by m, in the ratio 1 : 1 — m*. A second
operation of the same kind will reduce the amplitude to
(1 — m) 2 , and so on. After x passages the amplitude is
(1 — m) x , which if m be very small may be equated to e~ mx .
Here mx denotes the whole opaque area passed, reckoned
per unit area of wavefront ; and it would seem that the result
is applicable to any sufficiently sparse random distribution of
obstacles.
It may be of interest to give a numerical example. If the
unit of length be the centimetre and x the distance travelled,
m will denote the projected area of the drops situated in one
cubic centimetre. Suppose now that a is the radius of a
drop, and n the number of drops per cubic centimetre, then
m = mra 2 . The distance required to reduce the amplitude in
the ratio e : 1 is given by
x = 1/mra 2 .
Suppose that a=^ centim., then the abovenamed reduction
will occur in a distance of one kilometre (x = 90 b ) when n is
about 10 3 , i. e. when there is about one drop of one milli
metre diameter per litre.
It should be noticed that according to this theory a distant
point of light seen through a shower of rain ultimately become*!
invisible, not by failure of definition, but by loss of intensity
either absolutely or relatively to the scattered light.
* The intemity of the direct wave is 1 — 2?n, and that of the scattered
light m, making altogether 1—m.
[ 385 ]
XXXV. On Opacity, By Professor Oliver Lodge, D.Sc,
LL.D., PM.S., President of the Physical Society*.
MY attention has recently been called to the subject of the
transmission of electromagnetic waves by conducting
dielectrics — in other words, to the opacity of imperfectly
conducting material to light. The question arose when an
attempt was being made to signal inductively through a
stratum of earth or sea, how far the intervening layers of
moderately conducting material were able to act as a screen ;
the question also arises in the transmission of Hertz waves
through partial conductors, and again in the transparency of
goldleaf and other homogeneous substances to light.
The earliest treatment of such subjects is due of course to
Clerk Maxwell thirtyfour years ago, when, with unexampled
genius, he laid down the fundamental laws for the propagation
of electric waves in simple dielectrics, in crystalline media, and
in conducting media. He also realised there was some strong
aualogy between the transmission of such waves through space
and the transmission of pulses of current along a telegraph
wire. But naturally at that early date not every detail of the
investigation was equally satisfactory and complete.
Since that time, and using Maxwell as a basis, several
mathematicians have developed the theory further, and no
one with more comprehensive thoroughness than Mr. Oliver
Heaviside, who, as I have said before, has gone into these
matters with extraordinarily clear and far vision. I may
take the opportunity of calling or recalling to the notice of
the Society, as well as of myself, some of the simpler develop
ments of Mr. Heaviside's theory and manner of unifying
phenomena and processes at first sight apparently different ;
but first I will deal with the betterknown aspects of the
subject.
Maxwell deals with the relation between conductivity and
opacity in his Art. 7b>8 and on practically to the end of that
famous chapter xx , (' Electromagnetic Theory of Light ' ) . He
discriminates, though not very explicitly or obtrusively, be
tween the two extreme cases, (1) when inductive capacity or
electric inductivity is the dominant feature of the medium —
when, for instance, it is a slightly conducting dielectric, and
(2) the other extreme case, when conductivity is the pre
dominant feature.
* Communicated by the Physical Society of London, being the Presi
dential Address for 1899.
386 Dr. Oliver Lodge on Opacity.
The equation for the second case, that of predominant con
ductivity, is
da? " <r dt> U
F being practically any vector representing the amplitude of
the disturbance ; for since we need not trouble ourselves with
geometrical considerations such as the oblique incidence of
waves on a boundary &c, we are at liberty to write the y
merely as d/dx, taking the beam parallel and the incidence
normal.
No examples are given by Maxwell of the solution of this
equation, because it is obviously analogous to the ordinary
heat diffusion fully treated by Fourier.
Suffice it for us to say that, taking F at the origin as
represented by a simple harmonic disturbance Y =e ipt . the
solution of equation (1)
f? = ^?F (!')
dx 2 a
is F = F e~^ = 0Q*+fc«,
where Q = ^/(M) = ^/&. (1 + i , ;
wherefore
F = r(^)Sos(^(^J,), ... (2)
an equation which exhibits no true elastic wave propa
gation at a definite velocity, but a trailing and distorted
progress, with every harmonic constituent going at a diffe
rent pace, and dying out at a different rate ; in other words,
the diffusion so well known in the case of the variable stage of
heatconduction through a slab.
In such conduction the gain of heat by any element whose
heat capacity is cpdx is proportional to the difference of the
temperature gradient at its fore and aft surfaces, so that
, dB , _ dO
r dt dx
or, what is the same thing,
(Pd^cpdd
da? Tdt>
the same as the equation (1) above ; wherefore the constant
cp/k, the reciprocal of the thermometric conductivity, takes the
Dr. Oliver Lodge on Opacity. 387
place of 4:7t/jl/o; that is, of electric conductivity; otherwise the
heat solution is the same as (2). The 47T has come in from
an unfortunate convention, but it is remarkable that the con
ductivity term is inverted. The reason of the inversion of
this constant is that, whereas the substance conveys the heat
waves, and by its conductivity aids their advance, the aether
conveys the electric waves, and the substance only screens
and opposes, reflects, or dissipates them.
This is the case applied to seawater and low frequency by
Mr. Whitehead in a paper which he gave to this Society in
June 1897, being prompted thereto by the difficulty which
Mr. Evershed and the Post Office had found in some trials of
induction signalling at the Goodwin Sands between a coil
round a ship at the surface and another coil submerged at a
depth of 10 or 12 fathoms. It was suspected that the con
ductivity of the water mopped up a considerable proportion
of the induced currents, and Mr. Whitehead's calculation
tended, or was held to tend, to support that conclusion.
To the discussion Mr. Heaviside communicated what was
apparently, as reported, a brief statement ; but I learn that in
reality it w T as a carefully written note of three pages, which
recently he has been good enough to lend me a copy of. In
that note he calls attention to a theory of the whole subject
which in 1887 he had w T orked out and printed in his collected
' Electrical Papers,' but which has very likely been over
looked. It seems to me a pity that a note by Mr. Heaviside
should have been so abridged in the reported discussion as
to be practically useless ; and I am permitted to quote it here
as an appendix (p. 113).
Meanwhile, taking the diffusion case as applicable to sea
water with moderately low acoustic frequency;, we see that the
induction effect decreases geometrically with the thickness of
the oceanic layer, and that the logarithmic decrement of the
amplitude of the oscillation is \/( )> where a is the
specific resistance of seawater and pftir is the frequency.
Mr. Evershed has measured a and found it 2 x 10 10 C.G.S.,
that is to say 2 x 10 10 fx square centim. per second ; so putting
in this value and taking a frequency of 16 per second, the
amplitude is reduced to 1/eth of what its value would have
been at the same distance in a perfect insulator, by a depth
/ a /( 2 x 10 1( y \ _ /1(P_10 5
V 2*/*p V V2irii, x 2tt x 16 / " V 320 18 centllu 
= 55 metres.
Four or five times this thickness of intervening sea would
388 Dr. Oliver Lodge on Opacity.
reduce the result at the 16 frequency to insignificance (each
55metrelayer reducing the energy to \ of what entered it) ;
but if the frequency were, say, 400 per second instead of 16
it would be five times more damped, and the damping thick
ness (the depth reducing the amplitude in the ratio e : 1)
would in that case be only eleven metres.
It is clear that in a sea 10 fathoms (or say 20 metres)
deep the failure to inductively operate a "call " responding to
a frequency of 1 6 per second was not due to the screening effect
of seawater *.
Maxwell, however, is more interested in the propagation
of actual light, that is to say, in waves whose frequency is
about 5 x 10 u per second ; and for that he evidently does
not consider that the simple diffusion theory is suitable. It
certainly is not applicable to light passing through so feeble
a conductor as salt water. He attends mainly therefore to
the other and more interesting case, where electric inductive
capacity predominates over the damping effect of conduc
tivity, and where true waves therefore advance with an
approximately definite velocity
though it is to be noted that the slight sorting out of waves
of different frequency, called dispersion, is an approximation
to the case of pure diffusion where the speed is as the square
root of the frequency, and is accompanied, moreover, as it
ought to be, by a certain amount of differential or selective
absorption.
To treat the case of waves in a conductor, the same damping
term as before has to be added to the ordinary wave equation,
and so we have
£*£+?£ m
Taking ¥ =e ipt again, it may be written
g=(^ + ^)F, .... (*)
the same form as equation (1 ; ) ; so the solution is again
Y = e Q*+ipt }
* I learn that the ship supporting the secondary cable was of metal,
and that the primary or submerged cable was sheathed in uninsulated
metal, viz. in iron, which would no doubt be practically shortcircuited
by the seawater. Opacity of the medium is in that case a superfluous
explanation of the failure, since a closed secondary existed close to both
sending and receiving circuit.
Dr. Oliver Lodge on Opacity. 389
with Q 2 equal to the coefficient of F in (3') . Maxwell, however,
does not happen to extract the square root of this quantity,
but, assuming the answer to be of the form (for a simply
harmonic disturbance) [modifying his letters, vol. ii. § 798]
e~™ cos (pt — qx),
he differentiates and equates coefficients, thus getting
q 2 — r 2 = fJbKp 2 , 2rq= — ,
as the conditions enabling it to satisfy the differential equa
tion. This of course gives for the logarithmic decrement,
or coefficient of absorption,
2jrji .p
? *
p/q being precisely the velocity of propagation of the train of
waves. Though not exactly equal to 1/ V/llK, the true velo
city of wave propagation, except as a first approximation, in
an absorbing medium, yet practically this velocity p/q or X/T
is independent of the frequency except in strongly absorbent
substances where there are dispersional complications ; and
so the damping is_, in simple cases, practically independent
of the frequency too.
With this simple velocity in mind Maxwell proceeds to
apply his theory numerically to goldleaf, calculating its
theoretical transparency, and finding, as every one knows,
that it comes out discordant with experiment, being out of
all comparison * smaller than what experiment gives.
But then it is somewhat surprising to find gold treated as
a substance in which conductivity does not predominate over
specific inductive capacity.
The differential equation is quite general and applies to
any substance, and since the solution given is a true solu
tion, it too must apply to any substance when properly
interpreted ; but writing it in the form just given does not
suggest the full and complete solution. It seems to apply
only to slightly damped waves, and indeed, Maxwell seems to
consider it desirable to rewrite the original equation with omis
sion of K, for the purpose of dealing with good conductors.
By a slip, however, he treats gold for the moment as if it
belonged to the category of poor conductors, and as if ab
sorption in a thickness such as goldleaf could be treated as
a moderate damping of otherwise progressive waves.
* The fraction representing the calculated transmission by a film half
a wave thick has two thousand digits in its denominator : see below.
390 Dr. Oliver Lodge on Opacity.
The slip was naturally due to a consideration of the
extreme frequency of light vibrations ; but attention to the
more complete expression for the solution of the same differ
ential equation, given in 1887 by Mr. Heaviside and quoted
in the note to this Society above referred to, puts the matter
in a proper position. Referring to his ( Electrical Papers,'
vol. ii. p. 422, he writes down the general value of the
coefficient of absorption as follows (translating into our
notation)
rsMH&JT'}'
without regard to whether the conductivity of the medium is
large or small ; where v is the undamped or true velocity of
wave propagation in the medium ({aK)~K
Of course Maxwell could have got this expression in an instant
by extracting the square root of the quantity Q, the coefficient
of F in equation (o f ) written above. I do not suppose that
there is anything of the slightest interest from the mathe
matician's point of view, the interest lies in the physical
application ; but as this is not a mathematical Society it is
permissible, and I believe proper, to indicate steps for the
working out of the general solution of equation (3) by extract
ing the square root of the complex quantity Q.
The equation is
and the solution is
where
Q = ^/^K/ + ^^ =* + ;/? say.
Squaring we get, just as Maxwell did,
Squaring again and adding
(a* + f3 2 ) 2 = (a 2  /3 2 ) 2 + 4a 2 /3 2 = fi 2 Ky + ]^fjpt
I
wherefore
^v{. + (^)*}!
2/ 3.= P K i ,.{ v /(l+(i^)*) + l}, . . (4)
01
Dr. Oliver Lodge on Opacity. 391
and 2a 2 = the same with the last sign negative,
^(^[{i+^yyi]*. . . ( 5)
which is the logarithmic decrement of the oscillation per
unit of distance, or the reciprocal of the thickness which
reduces the amplitude in the ratio 1 : e (or the energy to f)
of the value it would have at the same place without
damping.
Using these values for a and ft, the radiationvector in
general, after passing through any thickness x of any medium
whose magnetic permeability and other properties are con
stant, is
F = F £ aX cos(^ / &z'), (6)
the speed of advance of the wavetrain being pi ft.
Now not only the numerical, value but the form of this
damping constant a, depends on the magnitude of the nume
4:7T
rical quantity — t?, which may be called the critical number*,
and may also be written
p*K/K > ••••••• A<)
where K, the absolute specific inductive capacity of the
medium, is replaced by its relative value in terms of K for
vacuum, and by — .=the velocity of light in vacuo =v .
vKoft
Now for all ordinary frequencies and good conductors this
critical number is large ; and in that case it will be found that
and that ft is identically the same. This represents the
simple diffusion case, and leads to equation (2).
On the other hand, for luminous frequency and bad con
ductors, the critical quantity is small, and in that case
* An instructive mode of writing a and /3 in general is given in (11")
or (12") below, where the above critical number is called tan e : —
av Vcose = p sine,
fiv Vcos e =■ p cos y.
392 Dr. Oliver Lodge on Opacity.
while
giving the solution
F=F ,^cos j p(j^). . . (8)
This expresses the transmission of light through imperfect
insulators, and is the case specially applied by Maxwell to cal
culations of opacity. Its form serves likewise for telegraphic
signals or Hertz waves transmitted by a highlyconducting
aerial wire ; the damping, if any, is independent of frequency
and there is true undistorted wavepropagation at velocity
v = 1/ VLB ; the constants belonging to unit length of the wire.
The current (or potential) at any time and place is
iu —
C = C e2Lv cosp(t— x vLS). ... (9)
The other extreme case, that of diffusion, represented by
equation (2), is analogous to the wellknown transmission of
slow signals by Atlantic cables, that is by long cables where
resistance and capacity are predominant, giving the socalled
KR law (only that I will write it RS),
C = C oe  7(l?,RS)x cos{^v(^RS) ( r}; . . (10)
wherefore the damping distance in a cable is
#Q
\/Grs)
Thus, in comparing the cable case with the penetration of
waves into a conductor and with the case of thermal con
duction, the following quantities correspond :
2W 2o ? 2k'
cp is the heatcapacity per unit volume, S is the electric
capacity per unit length ; k is the thermal conductivity per
unit volume, 1/R is the electric conductance per unit length.
So these agree exactly ; but in the middle case, that of waves
entering a conductor, there is a notable inversion, representing
a real physical fact. 4z7Tfju may be called the density and may
be compared with p or with 1/S, that is with elasticityhv 2 ;
but a is the resistance per unit volume instead of the con
ductance. The reason of course is that whereas good con
ductivity helps the cablesignals or the heat along, it by no
means helps the waves into the conductor. Conductivity aids
Dr. Oliver Lodge on Opacity* 393
their slipping along the boundary of a conductor, but it
retards their passing across the boundary and entering a
conductor. As regards waves entering a conductor, the
effect of conductivity is a screening effect, not a trans
mitting effect, and it is the bad conductor which alone has
a chance of being a transparent medium.
It may be convenient to telegraphists, accustomed to
think in terms of the " KRlaw " and comparing equa
tions (2) and (10), to note that the quantity 4c7t/jl/<t — that
is, practically, the specific conductivity in electromagnetic
measure (multiplied by a meaningless 47T because of an
unfortunate initial convention) — takes the place of KR (i. e. of
RS), but that otherwise the dampingout of the waves as
they enter a good conductor is exactly like the dampingout of
the signals as they progress through a cable ; or again as elec
trification travels along a cotton thread, or as a temperature
pulse makes its way through a slab ; and yet another case,
though it is different in many respects, yet has some simi
larities, viz. the ultimate distance the meltingpoint of wax
travels along a bar in Ingenhousz's conductivity apparatus, — 
the same law of inverse square of distance for effective reach
of signal holding in each case.
Now it is pointed out by Mr. Heaviside in several places
in his writings that, whereas the transmission of high
frequency waves by a nearly transparent substance corre
sponds by analogy to the conveyance of Hertz waves along
aerial wires (or along cables for that matter, if sufficiently con
ducting) , and whereas the absorption of lowfrequency waves
by a conducting substance corresponds, also by analogy, to the
diffusion of pulses along a telegraphcable whose selfinduction
is neglected — its resistance and capacity being prominent, —
the intermediate case of waves of moderate frequency in a
conductor of intermediate opacity corresponds to the more
general cable case where selfinduction becomes important and
where leakage also must be taken into account ; because it is
leakage conductance that is the conductance of the dielectric
concerned in plane waves. This last is therefore a real, and
not only an analogic, correspondence.
Writing R : Si L : Q 1 for the resistance, the capacity (" per
mittance"), the inductance, and the leakageconductance
(" leakance ") respectively, per unit length, the general
equations to cablesignalling are given in Mr. Heaviside's
Electromagnetic Theory ' thus : —
at ax at dec
394 Dr. Oliver Lodge on Opacity.
or for a simple harmonic disturbance,
g= (B,+yL0(Q 1 + vS,)V . . . (11)
= (*+WY,
whose solution therefore is
V = V e~ ax cos (ptpx) *...". (11')
There are several interesting special cases : —
The old cable theory of Lord Kelvin is obtained by omitting
both Q and L; thus getting equation (2).
The transmission of Hertz waves along a perfectlycon
ducting insulated wire is obtained by omitting Q and R ; the
speed of such transmission being 1/^(L 1 S 1 ). Resistance
in the wire brings it to the form (9), where the damping
depends on the ratio of the capacity constant RS to the self
induction constant L/S ; because the index R/2Li; equals half
the square root of this ratio ; but it must be remembered
that R has the throttled value due to merely superficial
penetration. The case is approximated to in telephony
sometimes.
A remarkable case of undistorted (though attenuated)
transmission through a cable (discovered by Mr. Heaviside,
but not yet practically applied) is obtained by taking
R/L = Q/S = r ;
the solution being then
rx / T \
▼•/HJ
due to fit) at #=0. All frequencies are thus treated alike,
and a true velocity of transmission makes its reappearance.
This is what he calls his ''distortionless circuit," which may
yet play an important part in practice.
And lastly, the two cases which for brevity may be
treated together, the case of perfect insulation, Q = 0, on
the one hand, and the case of perfect wire conduction, R=0,
on the other. For either of these cases the general expres
sion
.W =te .L4{ I + (f L )*}'{ 1+ (J)'}'{J?S'}]
* I don't know whether the following simple general expression for
a and /3 has been recorded by anyone : writing E/j9L=tan e and Q/^S =
tan e',
' p sin or cos K*+0 nt'n\
a or £ = L  . — 2 ' J , (11 )
v (cose cos e)i
which is shorter than (12).
(12)
Dr. Oliver Lodge on Opacity. 395
becomes exactly of the form (4) or (5) reckoned above for the
general screeningeffect, or opacity, of conducting media in
space.
For the number which takes the place of the quantity there
called the critical number, namely either R//>L or Q/pS, the
other being zero, we may write tan e ; in which case the
above is
«»ori8»=J>»L 1 S 1 (sece+l); (12')
or, rewriting in a sufficiently obvious manner, with 2tt/\ for
p/v if we choose,
_^sini6 pcos±e ^ ff
v{qos e)*' v(cose)a
Instead of attending to special cases, if we attend to the
general cable equation (11) as it stands, we see that it is
more general than the corresponding equation (3) to waves
in space, because it contains the extra possibility R of wire
resistance, which does not exist in free space.
Mr. Heaviside, however, prefers to unify the whole by the
introduction of a hypothetical and as yet undiscovered dissipa
tionpossibility in space, or in material bodies occupying
space, which he calls magnetic conductance, and which,
though supposed to be nonexistent, may perhaps conceivably
represent the reciprocal of some kind of hysteresis, either the
electric or the magnetic variety. Calling this g, (#H 2 is to
be the dissipation term corresponding with RC 2 ) , the equation
to waves in space becomes
V 2 F=(« ? + y. / a)(^ + ^K)F, . . . (13)
just like the general cable case. And a curious kind of
transparency, attenuation without distortion, would belong to
a medium in which both conductivities coexisted in such
proportion that g : ii = Airk : K ; for g would destroy H just
as k destroys E.
In the cable, F may be either current or potential, and
LSv 2 = l. In space, F may be either electric or magnetic
intensity, and /jlKv 2 =1 ; but observe that g takes the place
not of Q but of R, while it is Att/ct that takes the place of Q.
Resistance in the wire and electric conductivity in space
do not produce similar effects. If there is any analogue
in space to wire resistance it is magnetic not electric con
ductivity.
The important thing is of course that the wire does not
convey the energy but dissipates it, so that the dissipation by
wireresistance and the dissipation by spacehysteresis to
396 Dr. Oliver Lodge on Opacity.
that extent correspond. The screening effect of space
conductivity involves the very same dielectric property as that
which causes leakage or imperfect insulation of the cable
core.
Returning to the imaginary magnetic conductivity, let
us trace what its effects would be if it existed, and try to
grasp it. It effect would be to kill out the magnetism of
permanent magnets in time, and generally to waste away the
energy of a static magnetic field, just as resistance in w T ires
wastes the energy of an unmaintained current and so kills
out the magnetism of its field. I spoke above as if it were
conceivable that such magnetic conductivity could actually
in some degree exist, likening it to a kind of hysteresis ; but
hysteresis — the enclosure of a loop between a to and fro path
— is a phenomenon essentially associated with fluctuations, and
cannot exist in a steady field with everything stationary.
Admitted : but then the molecules are not stationary, and the
behaviour of molecules in the Zeeman and Righi phenomena,
or still more strikingly in the gratuitous radiations discovered
by Edmond Becquerel, and more widely recognized by others,
especially by Monsieur et Madame Curie, (not really gratuitous
but effected probably by conversion into highpitched radiation
of energy supplied from lowpitched sources), — the way
molecules of absorbent substances behave, seems to render
possible, or at least conceivable, something like a minute
magnetic conductivity in radiative or absorptive substances.
Mr. Heaviside, however, never introduced it as a physical
fact for which there was any experimental evidence, but as a
physical possibility and especially as a mathematical auxiliary
and unifier of treatment, and that is all that we need here
consider it to be ; but we may trace in rather more detail its
effect if it did exist.
Suppose the magnetism of a magnet decayed, what would
happen to its lines of force ? They would gradually shrink
into smaller loops and ultimately into molecular ones. The
generation of a magnetic field is always the opening out of
previously existing molecular magnetic loops ; there is no
such thing as the creation of a magnetic field, except in the
sense of moving it into a fresh place or expanding it over a
wider region *. So also the destruction of a magnetic field
merely means the shrinkage of its lines of force (or lines of
induction, I am not here discriminating between them).
Now consider an electric current in a wire : — a cylindrical
magnetic field surrounds it, and if the current gradually de
creases in strength the magnetic energy gradually sinks into
* This may be disagreed with.
Dr. Oliver Lodge on Opacity. 397
the wire as its lines slowly collapse. But observe that the
electric energy of the field remains unchanged by thi3
process : if the wire were electrostatically charged it would
remain charged, its average potential can remain constant.
Let the wire for instance be perfectly conducting, then the
current needs no maintenance, the potential might be
uniform (though in general there would be waves running to
and fro), and both the electric and magnetic fields continue
for ever, unless there is some dissipative property in space.
Two kinds of dissipative property may be imagined in
matter filling space : first, and most ordinary, an electric con
ductivity or simple leakage, the result of which will be to
equalize the potential throughout space and destroy the electric
field, without necessarily affecting the magnetic Held, and so
without stopping the steady circulation of the current mani
fested by that field. The other dissipativo property in space
that could be imagined would be magnetic conductivity ; the
result of which would be to shrink all the circular lines of
magnetic force slowly upon the wire, thus destroying the
magnetic field, and with it (by the circuital relation) the
current ; but leaving the electrostatic potential and the electric
field unchanged. And this imaginary effect of the medium
in surrounding space is exactly the real effect caused by what
is called electric resistance in the wire *.
Now for a simply progressive undistorted wave, i. e. one
with no character of diffusion about it, but all frequencies
travelling at the same quite definite speed l/\/yu,K,it is essential
that the electric and magnetic energies shall be equal. If
both are weakened in the same proportion, the waveenergy
is diminished, and the pulse is said to be " attenuated," but it
continues otherwise uninjured and arrives " undistorted/' that
is, with all its features intact and at the same speed as before,
but on a reduced scale in point of size.
This is the case of Mr. Heaviside's "distortionless circuit"
spoken of above, and its practical realization in cables, though
it would not at once mean Atlantic telephony, would mean
greatly improved signalling, and probably telephony through
shorter cables. In a cable the length of the Atlantic the
attenuation would be excessive, unless the absence of distortion
were secured by increasing rather the wire conductance than
the dielectric leakage ; but, unless excessive, simple attenua
* There is this difference, that in the real case the heat of dissipation
appears locally in the wire, whereas in the imaginary case it appears
throughout the magnetically conducting medium ; but I apprehend that
in the imaginary case the lines would still shrink, by reason of molecular
loops being pinched off them.
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 E
398 Dr. Oliver Lodge on Opacity.
tion does no serious harm. Articulation depends on the
features of the wave, and the preservation of the features
demands, by Fourier's analysis, the transmission of every
frequency at the same rate.
But now suppose any cause diminishes one of the two
fields without diminishing the other : for instance, let the
electric field be weakened by leakage alone, or let the mag
netic field be weakened by wireresistance alone, then what
happens ? The preservation of E and the diminution of H,
to take the latter — the ordinary — case, may be regarded as
a superposition on the advancing wave of a gradually growing
reverse field of intensity SH ; and, by the relation E = /LtuH,
this reversed field, for whatever it is worth, must mean a
gradually growing wave travelling in the reverse direction.
The ordinary wave is now no longer left alone and un
injured, it has superposed upon itself a more or less strong
reflected wave, a reflected wave which constantly increases
in intensity as the distance along the cable, or the penetration
of the wave into a conducting medium, increases ; all the
elementary reflected waves get mixed up by rereflexion in
the rear, constituting what Mr. Heaviside calls a diffusive
"tail" ; and this accumulation of reflected waves it is which
constitutes what is known as " distortion " in cables, and
what is known as "opacity" inside conducting dielectrics.
There is another kind of opacity, a kind due to hetero
geneousness, not connected with conductivity h\x' v due merely
to a change in the constants K and //,, — properly a kind of
translucency, a scattering but not a dissipation of energy, —
like the opacity of foam or ground glass.
This kind of opacity is an affair of boundaries and not of
the medium itself, but after all, as we now see, it has features
by no means altogether dissimilar to the truer kind of opacity.
Conducting opacity is due to reflexion, translucent opacity is
due to reflexion, — to irregular reflexion as it is called, but of
course there is nothing irregular about the reflexion, it is
only the distribution of boundaries which is complicated, the
reflexion is as simple as ever ; — except, indeed, to some
extent when the size of the scattering particles has to be
taken into account and the blue of the sky emerges. But
my point is that this kind of opacity also is after all of the
reflexion kind, and the gradual destruction of the advancing
Avave — whether it be by dust in the air or, as Lord Bayleigh
now suggests, perhaps by the discrete molecules themselves,
by the same molecular property as causes refraction and dis
persion — must result in a minute distortion and a mode of
wave propagation not wholly different from cablesignalling
Dr. Oliver Lodge on Opacity. 399
or from the transmission of lioht through conductors. So that
the red of the sunset sky and the green of goldleaf may not be
after all very different ; nor is the arrivalcurve of a telegraph
signal a wholly distinct phenomenon.
There is a third kind of opacity, that of lampblack, where
the molecules appear to take up the energy direct, converting
it into their own motion, that is into heat, and where there
appears to be little or nothing of the nature of reflexion. I
am not prepared to discuss that kind at present.
It is interesting to note that in the most resisting and
capacious cable that ever was made, where all the features of
every wave arrive as obliterated as if one were trying to
sigual by heatpulses through a slab, that even there the head
of every wave travels imdistorted, with the velocity of light,
and suffers nothing but attenuation ; for the superposed re
versed field is only called out by the arrival of the direct pulse,
and never absolutely reaches the strength of the direct field.
The attenuation may be excessive, but the signal is there
in its right time if only we have a sensitive enough instru
ment to detect it ; though it would be practically useless as a
signal in so extreme a case, being practically all tail.
Nothing at all reaches the distant end till the lightspeed
time has elapsed ; and the lightspeedtime in a cable depends
on the /jl and K of its insulating sheath, depends, if that is
not simply cylindrical, on the product of its selfinductance and
capacity per unit length ; but at the expiration of the light
speedtime the head of the signalling pulse arrives, and
neither wireresistance nor insulationleakage, no, nor mag
neticconductivity, can do anything either to retard it or to
injure its sharpness : they can only enfeeble its strength, but
they can do that very effectually.
The transmitter of the pulse is selfinduction in conjunction
with capacity : the chief practical enfeebler of the pulse
is wireresistance in conjunction with capacity ; and before
Atlantic telephony is possible (unless a really distortionless
cable is forthcoming) the copper core of an ordinary cable
will have to be made much larger. Nothing more is wanted
in order that telephony to America may be achieved. There
may be practical difficulties connected with the mechanical
stiffness of a stout core and the worrying of its guttapercha
sheath, and these difficulties may have to be lessened by aiming
at distortionless conditions — it is well known also that for
high frequencies a stout core must be composed of insulated
strands unless it is hollow — but when such telephony is accom
plished, I hope it will be recollected that the full and complete
principles of it and of a great deal else connected with tele
2E 2
400 Dr. Oliver Lodge on Opacity.
graphy have been elaborately and thoroughly laid down by
Mr. Heaviside.
There is a paragraph in Maxwell, concerning the way
a current rises in a conductor and affects the surrounding
space, which is by no means satisfactory : it is Art. 804.
He takes the current as starting all along the wire, setting up
a sheath of opposition induced currents in the surrounding
imperfectly insulating dielectric, which gradually diffuse out
wards and die away, leaving at last the full inductive effect
of the corecurrent to be felt at a distance. Thus there is
supposed to be a diffusion of energy outwards from the wire,
which he likens to the diffusion of heat.
But, as Mr. Heaviside has shown, the true phenomenon is
the transmission of a wave in the space surrounding the
wire — a plane wave if the wire is perfectly conducting,
a slightly coned wave if it resists, — a wavefront perpendicular
to the wire and travelling along it, — a sort of beam of dark
light with the wire as its core.
Telegraphic signalling and optical signalling are similar ;
but whereas the beam of the heliograph is abandoned to
space and must go straight except for reflexion and refraction,
the telegraphic beam can follow the sinuosities of the wire and
be guided to its destination.
If the medium conducts slightly it will be dissipated
in situ; but if the wire conducts imperfectly, a minute trickle
of energy is constantly directed inwards radially towards the
wire core, there to be dissipated as heat. Parallel to the
wire flows the main energy stream, but there is a small
amount of tangential grazing and inward flow. The initial
phenomenon does not occur in the wire, gradually to spread
outwards, but it occurs in the surrounding medium, and a
fraction of it gradually converges inwards. The advancing
waves are not cylindrical but plane weaves, and though the
diffusing waves are cylindrical they advance inwards, not
outwards.
I will quote from a letter of Mr. Heaviside's: — " The easiest
way to make people understand is, perhaps, to start with a
conducting dielectric with plane waves in it without wires
[thus getting] one kind of attenuation and distortion. Then
introduce wires of no resistance ; there is no difference except
in the v 7 ay the lines of force distribute [enabling the wires to
guide the plane waves]. Then introduce magnetic con
ductivity in the medium, [thereby getting] the other kind of
attenuation and distortion. Transfer it to the wires, makino
it electrical resistance. Then abolish the first electric con
ductivity, and you have the usual electric telegraph,"
Dr. Oliver Lodge on Opacity, 401
Opacity of Goldleaf.
Now returning to the general solution (5) let us apply it
to calculate the opacity of goldleaf to light.
Take cr = 2000fJL square centim. per sec,
p = 2irX 5 X 10 14 per sec. ;
then the critical quantity 47r/pcrK or (7) is
2x9 xlO 20 1800
5xl0 14 x2000K/K K/K *
This number is probably considerably bigger than unity
(unless, indeed, the specific inductive capacity K/K of gold
is immensely large, which may indeed be the case — refractive
index 40, for instance, — only it becomes rather difficult to
define) ; so that, approximately,
// 27rup \ /40 x 5 x 10 u ,. n „ . n R
« = V (jF) = V 2000 = ^ 10 = 3 x 106 ;
or the damping distance is
F x x 10~ 5 centim. = j microcentimetre,
whereas the wavelength in air is
6 x 10~ 5 centim. = 60 microcentimetres.
The damping distance is therefore getting nearer to the
right order of magnitude, but the opacity is still excessive.
A common thickness for goldleaf is stated to be half a
wavelength of light ; that is to say, 90 times the damping
distance. Hence the amplitude of the light which gets
through a half wave thickness of gold is e~ 9o of that which
enters ; and that is sheer opacity.
[Maxwell's calculation in Art. 798, carried out numerically,
makes the damping
_ 2tt\xg
e * x , = exp. (— >10 8 #) for gold,
see equation (8) above ; or, for a thickness of half a wave
length, 10 1000 , which is billions of billions of billions (indeed
a number with 960 digits) times greater opacity than what
we have here calculated, and is certainly wrong.]
It must, however, be granted, I think, that the green light
that emerges from goldleaf is not properly transmitted ; it
is light reemitted by the gold *. The incident light, say the
* This would be fluorescence, of course ; and Dr. Larmor argues in
favour of a simple ordinary exponential coefficient of absorption even in
metals. See Phil. Trans. 1894, p. 738, § 27.
402 Dr. Oliver Lodge on Opacity.
red, is all stopped by a thickness less than half a wavelength.
The green light may conceivably be due to atoms vibrating
fairly in concordance, and not calling out the conducting
opacity of the metal. If the calculated opacity, notwith
standing this, is still too great, it is no use assuming a higher
conductivity at higher frequency, for that would act the
wrong way. What must be assumed is either some special
molecular dispersion theory, or else greater specific resistance
for oscillations of the frequency which get through ; nor
must the imaginative suggestion made immediately below
equation (13) be altogether lost sight of.
There is, however, the possibility mentioned above that the
relative specific inductive capacity of gold, K/K , if a
meaning can be attached to it, may be very large, perhaps
(though very improbably, see Drude, Wied. Ann. vol. xxxix.
p. 481) comparable with 1800. Suppose for a moment that
it is equal to 1800 ; then the value of the critical quantity (7)
is 1 and the value of a is
= 19xl0 5 ,
which reduces the calculated opacity considerably, though
still not enough.
In general, calling K/K = r, and writing the critical
number — — as h/c. we have
ape
% a 2 _ I a 2 //3 2 = XV/27T 2 = ^ (7j2 _j_ c 2) _ c .
so that aX/'Iir ranges from s/ ^1l when h/c is big, to \h/^ c
when h/c is small.
Writing the critical number h/c as tan e, the general value
of a is given by
aX=7r \/2c(sece — 1) (14)
This is the ratio of the wavelength in air to the damping
distance in the material in general ; meaning by " the damp
ing distance " the thickness which reduces the amplitude in
the ratio e : 1. (14) represents expression (5) ; compare
with (120
Theory of a Film.
So far nothing has been said about the limitation of the
medium in space, or the effect of a boundary, but quite
recently Mr. Heaviside has called my attention to a special
Dr. Oliver Lodge on Opacity. 403
theory, a sort of Fresnellike theory, which he has given
for infinitely thin films of finite conductance ; it is of remark
able simplicity, and may give results more in accordance
with experiment than the theory of the universal opaque
medium without boundary, hitherto treated : a medium in
which really the source is immersed.
Let a film, not so thick as goldleaf, but as thin as the
black spot of a soapbubble, be interposed perpendicularly
between source and receiver. I will quote from i Electrical
Papers,' vol. ii. p. 385 : — " Let a plane wave ^j 1 =:/jlvH 1
moving in a nonconducting dielectric strike flush an ex
ceedingly thin sheet of metal [so thin as to escape the need
for attending to internal reflexions, or the double boundary,
or the behaviour inside] ; letE 2 = //uH 2 be the transmitted wave
out in the dielectric on the other side, and E 3 = — /jlvH 3 be the
reflected wave *.
* General Principles. — It may be convenient to explain here the
principles on which Mr. Heaviside arrives at his remarkably neat
expression for a wavefront in an insulating medium,
E = pvH,
or as it may be more fully and vectorially written,
VOE) = ^H,
where E is a vector representing the electric intensity (proportional to
the electric displacement), H is the magnetic intensity, and v is unit
normal to the wavefront. E and H are perpendicular vectors in the
same plane, i. e. in the same phase, and E H v are all at right angles to
each other.
The general electromagnetic equations in an insulating medium are
perhaps sufficiently well known to be, on Mr. Heaviside's system,
curl K = KE and —curl E = ^H,
where " curl " is the vector part of the operator v, and where Maxwell's
vectorpotential and other complexities have been dispensed with.
[In case these equations are not familiar to students I interpolate a
parenthetical explanation which may be utilised or skipped at pleasure.
The orthodox definition of Maxwell's name " curl " is that b is called
the curl of a when the surfaceintegral of ft through an area is equal to the
lineintegral of a round its boundary, a being a vector or a component
of a vector agreeing everywhere with the boundary in direction, and b
being a vector or component of vector everywhere normal to the area.
Thus it is an operator appropriate to a pair of looped or interlocked
circuits, such as the electric and the magnetic circuits alwa}s are. The
first of the above fundamental equations represents the fact of electro
magnetism, specially as caused by displacement currents in an insulato?',
the second represents the fact of magnetoelectricity, Faraday's magneto
electric induction, in any medium. Taking the second first, it states the
fundamental law that the induced EMF in a boundary equals the rate of
change in the lines of force passing through it ; since the EMF or step
of potential all round a contour is the lineintegral of the electric intensity
E round it, so that
EMF = f Eds =  ^=fTBe?S=ff«HrfS :
J cycle at J J JJ
404 Dr. Oliver Lodge on Opacity.
" At the sheet we have
Ei + E 3 = E 2
H 1 +H 3 =H 2 +47r^E 2 ,
k being the conductivity of the sheet of thickness z. Therefore
Eg __ H 2 Ej f E 3 __ 1
E x H!  E7~ s= l+2«/i*8w' '
(15)
wherefore — ( t«H equals the curl of E. (The statement of this second
circuital law is entirely due to Mr. Heaviside ; it is now largely adopted
and greatly simplifies Maxwell's treatment, abolishing the * need for
vector potential.)
The first of the above two fundamental equations, on the other hand,
depends on the fact that a current round a contour excites lines of mag
netic force through the area bounded by it, and states the law that the
total magnetomotive force, or lineintegral of the magnetic intensity
round the boundary, is equal to 4n times the total current through it j
the total current being the " ampereturns " of the practical Engineer.
Expressing this law in terms of current density c, we write
MMF = f Hds = 4ttC = ff 4*cd$
J cycle J.'
so always currentdensity represents the curl of the magnetic field due
to it, or curl H=47rc.
Now in a conductor c = kE, but in an insulator c = D, the rate of change
of displacement or Maxwell's " displacementcurrent " ; and the dis
placement itself is proportional to the intensity of the electric field,
D = — E ; hence the value of current density in general is
47T
c=*E+*E,
tt7T
whence in general
curl H = 47r&E+KE = (4ttA'+K/?)E,
and in an insulator the conductivity k is nothing.
The connexion between " curl " so defined and Vv is explained as
follows. The operator v applied to a vector R whose components are
X Y Z gives
. d , .. d , 7 . d
~dy
which, worked out, yields two parts
Q /dX,dY.dZ\
also called convergence, and
T7 ./ dZ dY\ , ■ /dX dZ \ , , laY dX\
or say « \jt] + Jc£, where £ rj £ are the components of a spinlike
vector <». Now a theorem of Sir George Stokes shows that the normal
component of co integrated over any area is equal to the tangential com
ponent of R integrated all round its boundary ; hence Vv and curl are
the same thing.
(•a +>s +*£)(*+.*+»>.
Dr. Oliver Lodge on Opacity, 405
H is reflected positively and E negatively. A perfectly con
ducting barrier is a perfect reflector ; it doubles the magnetic
force and destroys the electric force on the side containing
the incident wave, and transmits nothing/'
[I must here interpolate a remark to the effect that though
it can hardly be doubted that the above boundary conditions
(tangential continuity of both E and H) are correct, yet
in general we cannot avoid some form of sethertheory
Whenever ox is zero it follows that R has no circulation but is
the derivative of an ordinary singlevalued potential function, whose
dV =X.dx{ Ydy\Zdz. In electromagnetism this condition is by no means
satisfied. E and H or H and E are both full of circulation, and their
circuits are interlaced. Fluctuation in E by giving rise to current causes
H ; fluctuation in H causes induced E.]
Now differentiating only in a direction normal to a plane wave
advancing along a; the operator Vv becomes simply idjdx when applied
to any vector in the wavefront, the scalar part of v being nothing.
So the second of the above fundamental equations can be written
■ dE_ dH
~ l d^~^W
dE
so, ignoring any superposed constant fields of no radiation interest, E and
II are vectors in the same phase at right angles to each other, and their
tensors are given by E = pvYL.
Similarly of course the other equation furnishes H=KvE; thus giving
the ordinary K^v 2 = l, and likewise the fact that the electric and mag
netic energies per unit volume are equal, KE 2 =^H 2 .
A wave travelling in the opposite direction will be indicated by
E= pvH; hence, as is well known, if either the electric or the
magnetic disturbance is reversed in sign the direction of advance is
reversed too.
(The readiest way to justify the equation E=^wH, a posteriori, is to
assume the two wellknown facts obtained above, viz. that the electric
and magnetic energies are equal in a true advancing wave, and that
0=1/ >J pK; then it follows at once.)
Treatment of an insulating boundary. — At the boundary of a different
medium without conductivity the tangential continuity of E and of H
across the boundary gives us the equations
E, + E 3 = E 2
H 1 + H 3 =H 2 ,
where the suffix 1 refers to incident, the suflix 2 to transmitted, and
the suffix 3 to reflected waves.
H^rHg may be replaced by /»v(Ej  E 3 ), since the reflected wave is
reversed ; so we shall have, for the second" of the continuity equations,
^E. 2 =mnE 2 ;
406 Dr. Oliver Lodge on Opacity.
when we have to lay down continuity conditions, and,
according to the particular kind of aethertheory adopted so
will the boundary conditions differ. My present object is to
awaken a more general interest in the subject and to repre
sent Mr. Heaviside's treatment of a simple case ; but it must
be understood that the continuity conditions appropriate to
oblique incidence have been treated by other great mathe
matical physicists, notably by Drude, J. J. Thomson, and
Larmor, also by Lord Rayleigh, and it would greatly
enlarge the scope of this Address if I were to try to discusss
the difficult and sometimes controversial questions which
arise. I must be content to refer readers interested to the
writings of the Physicists quoted — especially I may refer to
J. J. Thomson's ' Recent Researches,' Arts. 352 to 409, and
to Larmor, Phil. Trans, 1895, vol. 186, Art, 30, and other
places.]
Now apply this to an example. Take k for gold, as we
have done before, to be 1/2000 fi seconds per square centim.
and v = Sx 10 10 centim. per sec, for v is the velocity in the
n being the index of refraction, and m the relative inductivity. Hence,
adding and subtracting,
E_ 2= 2
E, 1hW
and
E 3 \—nm m
E,~~ 1+nm '
wellknown optical expressions for the transmitted and reflected ampli
tudes at perpendicular incidence, except that the possible magnetic property
of a transparent medium is usually overlooked.
Treatment of a conducting boundary. — But now, if the medium on the
other side of the boundary is a conductor instead of a dielectric, a term
in one of the general equations must be modified; and, instead of
curl H=KjpE, Ave shall have, as the fundamental equation inside the
medium,
— — _ = 4ttA;L;
dx
or more generally (4ttZ;+K//)E.
So, on the far side of a thin slice of thickness z, the magnetic intensity
H 2 is not equal to the intensity H^+Hg on the near side, but is less by
f/H = 4:TrkEdx = 4:7rkE 2 z = 4nkftvzH. a ;
and this explains the second of the continuity equations immediately
following in the text.
In a quite general case, where all the possibilities of conductivity and
capacity &c. are introduced at once, the ratio of E/H is not pv or O/K)*?
but is ( y g^u.pf{4iiTk\\i.p)~ 2 for waves in a general material medium,
{g may always be put zero), or (RjpL)*(Q.+^S) 2 for waves guided by
a resisting wire through a leaky dielectric.
The addition of dielectric capacity to conductivity in a film is there
fore simple enough and results in an equation quoted in the text below.
Dr. Oliver Lodge on Opacity. 407
dielectric not in the conductor ; then take a film whose
thickness z is one twentyfifth of a wavelength of the
incident light ; and the ratio of the transmitted to the
incident amplitude comes out
1/9 T, 1000 ]
irvz 200
Some measurements made by W. Wien at Berlin in 1888
(Wied. Ann. vol. xxxv.), with a bunsenburner as source of
radiation, give as the actual proportion of the transmitted
to the longwave incident light, for gold whose thickness is
10 5 centim., '0033 or 1/300 ; while for gold one quarter as
thick the proportion was 0'4 (see Appendix II. page 414).
He tried also two intermediate thicknesses, and though
approximately the opacity increases with the square of the
thickness, it really seems to increase more rapidly : as no
doubt it ought, as the boundaries separate. However, for
a thickness X/25 I suppose we may assume that about l/3rd
of the light would be transmitted, whereas the filmtheory
Simple treatment of the E.M. theory of light. — It is tempting to show
bow rapidly the two fundamental electromagnetic equations, in Mr. Heavi
side's form, lead to the electromagnetic theory of light, if we attend
specially to the direction normal to the plane of the two perpendicular
vectors E and H, to the direction along say x, so that v = id/dx and
v 2 = djdx 2 .
In an insulating medium the equations are
curlH = KE and curlE=^H;
now curl=Vv = Vj since Sv = in this case, so
V 2 H=KvE=KcurlE= E>H;
or, in ordinary form,
<£H_ K . dm
dx 2 ** dt 2 *
and there are the waves.
If this is not rigorous, there is no difficulty in finding it done properly
in other places. I believe it to be desirable to realize things simply as
well.
In a conducting medium the fundamental equations are, one of them,
curl H = K E + 4tt&E = (Kp + ink) E,
while the other remains unchanged; unless we like to introduce the non
existent auxiliary g, which would make it
VvE=(<7+^)H,
and would cover wires too.
So  v 2 H= (4tt&+I^)(#+^)H,
the general wave equation. In all these equations p stands for djdt ; but,
for the special case of simply harmonic disturbance of frequency pl'2ir, of
course ip can be substituted.
408 Dr. Oliver Lodge on Opacity.
gives (1/200) 2 ; so even now a metal calculates out too opaque,
though it is rather less hopelessly discrepant than it used to
be. The result, we see, for the infinitely thin film, is inde
pendent of the frequency.
Specific inductive capacity has not been taken into account
in the metal, but if it is it does not improve matters. It
does not make much difference, unless very large, but what
difference it does make is in the direction of increasing
opacity. In a letter to me Mr. Heaviside gives for the
opacity of a film of highly conducting dielectric
^=£ L (l + 27r f JLkvz)* + (imcz I j/v) 2 } , . . (16)
where I have replaced his %fivzK.p last term by an expression
with the merely relative numbers K/K and /j,//j, , called c
and m respectively, thus making it easier to realise the
magnitude of the term, or to calculate it numerically.
Theory of a Slab.
An ordinary piece of goldleaf, however, cannot properly
be treated as an infinitely thin film ; it must be treated as a
slab, and reflexions at its boundaries must be attended to.
Take a slab between x = and x=l. The equations to be
satisfied inside it are the simplified forms of the general
fundamental ones
£»*; . § ME <
k being l/<r, and K being ignored ; while outside, at # = Z,
the condition E=yLtvH has to be satisfied, in order that a
wave may emerge.
The following solutions do all this if q 2 = ^irfxkp : —
V qv —p J
p \ qv—p J
Conditions for the continuity of both E and H at x =
suffice to determine A, namely if E x Hj is the incident and
E 3 H 3 the reflected wave on the entering side, while E H
are the values just inside, obtained by putting <r = in the
above,
Ei + E 3 = E ,
E 1 E 3 =^(H 1 + H 3 )=^H .
Adding, we get a value for A in terms of the incident light E l5
2E 1 p(qvp) = A{(qv+pye 2ql (qvp) 2 }.
Dr. Oliver Lodge on Opacity. 409
whence we can w T rite E anywhere in the slab,
1  Mw—p) f eq , . Q±±£ e 2 q i e  q A .
E i " (qv+py^iqvp)* \ * gvp J
Put x = l, and call the emergent light E 2 ; then
% ~ {qv+pye« l (gvp) 2 e* 1 " Pi ^ ' l }
and this constitutes the measure of the opacity of a slab,
p z being the proportion of incident light transmitted.
It is not a simple expression, because of course p signifies
the operator d/dt, and though it becomes simply ip for a
simply harmonic disturbance, yet that leaves q complex.
However, Mr. Heaviside has worked out a complete expres
sion for p 2 , which is too long to quote (he will no doubt be
publishing the whole thing himself before long), but for slabs
of considerable opacity, in which therefore multiple re
flexions may be neglected, the only important term is
4,s/2ey/
av
with
?= l + (l+p/*vf ••••• ( 18 )
* = 4/(2wpp/a) =3xlO G
for light in gold ; and
V 27T 27T 1 .
± — — — = 7777 — e = rrr about.
av a\ 60 x 5 25
So the effect of attending to reflexion at the walls of the slab
is to still further diminish the amplitude that gets through,
below the e~ al appropriate to the unbounded medium, in the
ratio of rtg , or about a ninth.
25 '
Effect of each Boundary,
It is interesting to apply Mr. Heaviside's theory to a study
of what happens at the first boundary alone, independent of
subsequent damping.
Inside the metal, by the two fundamental equations, we
have
and by continuity across the boundary
E 1 + E 3 =E ,
E 1 E 3 =„H =At ,E (^)WJ'E ,
where still q 2 = A7rfjbkp.
410 Dr. Oliver Lodge on Opacity.
Therefore,, for the transmitted amplitude
E _ 2p
and for the reflected
Ej p f gv
5_3 ^ p — qv
B x p + qv
or rationalising and writing amplitudes only, and under
standing by p no longer d/dt in general, but 'only 2tt times
the frequency,
Eo _ 2p
E, ^((p + av y + MP^' ■ ( u )
Any thickness of metal multiplies this by the factor e~ ax ,
and then comes the second boundary, which, according to
what has been done above, has a comparatively small but
peculiar effect ; for it ought to change the amplitude from p 1
into p, that is to give an emergent amplitude
_4/ 2. p/*v_
l + (l+p/avy
instead of the above incident on the second boundary
^ e,«i. am
v(i+(i+^M 3 ) ' " ' (0)
that is for the case of light in gold, for which p/av is small,
to change 2/ s/2 into 2 n/2, in other words, to double it.
2 /2tt
The effect of the first boundary alone, p 1} is —  — , or say
1/18, and this is a greater reduction effect than that reckoned
above for the two boundaries together.
Thus the obstructive effect of the two boundaries together
comes out less than that of the first boundary alone — an
apparently paradoxical result. About oneeighteenth of the
lightamplitude gets through the first boundary, but about
oneninth gets through the whole slab (ignoring the geo
metrically progressive decrease due to the thickness, that is
ignoring e~ al , and attending to the effect of the boundaries
alone ; which, however, cannot physically be done). At
first sight this was a preposterous and ludicrous result. The
second or outgoing boundary ejects from the medium nearly
double the amplitude falling upon it from inside the con
ductor ! But on writing this, in substance, to Mr. Heaviside
he sent all the needful answer by next post. "The incident
disturbance inside is not the whole disturbance inside."
Dr. Oliver Lodge on Opacity. 411
That explains the whole paradox — there is the reflected
beam to be considered too. At the entering boundary the
incident and reflected amplitudes are in opposite phase, and
nearly equal, and their algebraic sum, which is transmitted,
is small. At the emerging boundary the incident and re
flected amplitudes are in the same phase, and nearly equal,
and their algebraic sum, which is transmitted, is large — is
nearly double either of them. But it is a curious action : —
either more light is pushed out from the limiting boundary
of a conductor than reaches it inside, or else, 1 suppose,
the Telocity of light inside the metal must be greater
than it is outside, a result not contradicted by Kundt's
refraction experiments, and suggested by most optical
theories. It is worth writing out the slab theory a little
more fully, to make sure there is no mistake, though the
whole truth of the behaviour of bodies to light can hardly be
reached without a comprehensive molecular dispersion theory.
I do not think Mr. Heaviside has published his slab theory
anywhere yet. A slab theory is worked out by Prof. J. J.
Thomson in Proc. Roy. Soc. vol. xlv., but it has partly
for its object the discrimination between Maxwell's and other
rival theories, so it is not very simple. Lord Kelvin's Balti
more lectures probably contain a treatment of the matter.
All that I am doing, or think it necessary to do in an
Address, is to put in palatable form matter already to a few
leaders likely to be more or less known : in some cases
perhaps both known and objected to.
The optical fractions of Sir George Stokes, commonly
written h c e f, are defined, as everyone knows, as follows.
A ray falling upon a denser body with
incident amplitude 1 yields a reflected
amplitude h and a transmitted c. A
ray falling upon the boundary of a
rare body with incident amplitude 1
has an internally reflected amplitude e
and an emergent /. General prin
ciples of reversibility show that
b\e=0, and that b 2 + cf= 1 in a trans
parent medium.
Now in our present case we are attending to perpendicular
incidence only, and we are treating of a conducting slab;
indeed, we propose to consider the obstructive power of the
material of the slab so great that we need not suppose that
any appreciable fraction of light reflected at the second surface
returns to complicate matters at the first surface. This limita
tion by no means holds in Mr. Heaviside's complete theory, of
course, but 1 am taking a simple case.
412 Dr. Oliver Lodge on Opacity.
The characteristic number which governs the phenomenon
is — or — , a number which for light and gold we reckoned as
being about gL tnat is decidedly smaller than unity, a being
s /(2wfMkp) or a / (  77 ^ ). The characteristic number p/uv
we will for brevity write as h, and we will express amplitudes
for perpendicular incidence only, as follows : —
Incident amplitude 1,
externally reflected b= — < z — \. r^ \
i 1 + (I h/i) J
2h
entering
Incident again 1,
internally reflected e— < — * i '
2\/2
emergent /= {1 + (1 + fe)2 ^
(It must be remembered that e and / refer to the second
boundary alone, in accordance with the above diagram.)
Thus the amplitude transmitted by the whole slab, or
rather by both surfaces together, ignoring the opacity of its
material for a moment, is
transmitted cf= ., — / ., ,* 7 NO «
1 + ( 1 + A) 2
To replace in this the effect of the opaque material, of
thickness I, we have only to multiply by the appropriate ex
ponential damper, so that the amplitude ultimately trans
mitted by the slab is
4;\/2.p/uv _ al
l+(l+p/*v)* e a
times the amplitude originally incident on its front face.
This agrees with the expression (18) specifically obtained
above for this case, but, once more I repeat, multiple reflexions
have for simplicity been here ignored, and the medium has
been taken as highly conducting or very opaque.
But even so the result is interesting, especially the result
for /. To emphasize matters, we may take the extreme case
when the medium is so opaque that h is nearly zero ; then b
is nearly — 1, c is nearly 0, being hs/2, e is the same as h
except for sign, and /is nearly 2.
Dr. Oliver Lodge on Opacity. 413
An opaque slab transmits Sh 2 e~ 2al of the incident light
energy ; its first boundary transmits only 2nh 2 . The second
or emergent boundary doubles the amplitude. Taken in
connexion with the facts of selective absorption and the
timing of molecules to vibrations of certain frequency, I
think that this fact can hardly be without influence on the
green transparency of goldleaf.
Appendix I.
Mr. Heaviside's Note on Electrical Waves in Sea Water.
[Contributed to a discussion at the Physical Society in June 1897 :
see Mr. Whitehead's paper, Phil. Mag. August 1897.]
" To find the attenuation suffered by electrical waves through
the conductance of seawater, the first thing is to ascertain
whether, at the frequency proposed, the conductance is paramount,
or the permittance, or whether both must be counted.
"It is not necessary to investigate the problem for any particular
form of circuit from which the waves proceed. The attenuating
factor for plane waves, due to Maxwell, is sufficient. If its validity
be questioned for circuits in general, then it is enough to take the
case of a simplyperiodic point source in a conducting dielectric
(' Electrical Papers,' vol. ii. p. 422, § 29). The attenuating constant
is the same, viz. (equation (199) loc cit.) : —
where n/2ir is the frequency, Tc the conductivity, c the permittivity,
and v=(fxc)~i, fi being the inductivity.
" The attenuator is then e~ n \ r at distance r from the source, as
in plane waves, disregarding variations due to natural spreading.
It is thus proved for any circuit of moderate size compared with
the wavelength, from which simply periodic waves spread.
" The formula must be used in general, with the best values of h
and c procurable. But with long waves it is pretty certain that the
conductance is sufficient to make 47rl/cn large. Say with common
saltsolution ^ = (30 11 )~ 1 , then
4ttI: _ 2kjxv 2
en ~~ f
if /is the frequency. This is large unless /is large, whether we
assume the specific c/c to have the very large value 80 or the
smaller value effectively concerned with light waves. We then
reduce n x to
n 1 =(2n t xJc7rf=:2r(fx7cff,
as in a pure conductor.
" This is practically true perhaps even with Hertzian waves, of
which the attenuation has been measured in common saltsolution
by P. Zeeman. If then I— ] =30 n [and if the frequency is 300 per
second] we get n x = about ^nnr
Phil. Mag. S. 5. Vol. 47. No. 287. April 1899. 2 F
414
Dr.
Oliver Lodge on
Opacity.
" Therefore 50 metres is the distance in which the attenuation
due to conductivity is in the ratio 2*718 to 1, and there is no reason
why the conductivity of seawater should interfere, if the vahie is
like that assumed above.
"These formulae and results were communicated by me to
Prof. Ayrton at the beginning of last year, he having enquired
regarding the matter, on behalf of Mr. Evershed I believe.
" The doubtful point was the conductivity. I had no data, but
took the above I' from a paper which had just reached me from
3Ir. Zeeman. Xow Mr. "Whitehead uses fr 1 = 20 10 , which is no
less than 15 times as great. I presume there is good authority for
this datum *. Xone is given. Using it we obtain » 1 = 13 1 16 .
" Thus 50 metres is reduced to 1316 metres. But a considerably
greater conductivity is required before it can be accepted that the
statements which have appeared in the press, that the failure of
the experiments endeavouring to establish telegraphic communica
tion with a lightship from the seabottom was due to the con
ductance of the sea, are correct. It seems unlikely theoretically.
and Mr. Stevenson has contradicted it (in 'Nature') from the
practical point of view. So far as I know, no account has been.
published of these experiments, therefore there is no means ol
finding the cause of the failure."
Appendix II.
The experiments of W. Wien on the transparency of metals,
bv means of a bolometer arranged to receive the radiation from a
bunsen burner transmitted through different films, resulted in the
following numbers for the proportion of radiation transmitted.
Proportion transmitted.
Metal.
Thickness in
10~ : centim.
Bunsen burner
Bunsen burner
Proportion
reflected.
luminous.
nonluminous.
•13
Platinum
20
•32
•37
Iron & Platinum
40420
•10
■14
•45
Gold 1
56
•040
•041
•63
Gold2
100
24
35
36
•0035
•41
•20
•058
•0036
•41
•20
•046
•80
•05
■19
•78
Gold 3
Gold 4
Silver 1 (blue) ...
Silver 2 (grey) . . .
395
•058
•055
•60
Silver 3 (grey) . . .
29
•25
•42
•40
Silver 4 (blue) ...
597
•0022
•0019
•95
Silver 5 (grey) ...
273
•31
•43
•24
* Dr. J. L. Howard has recently set a student to determine the resistivity
of the seawater used by Professor Herdman, density 1019 gr. per c.c,
and he finds it to be 3x 10 10 c.g.s. at 15° C— O. J. L*, March 1899.
Prof. J. J. Thomson on ilie Cathode Rays. 415
The thickness is in millionths of a millimetre, i. e. is in terms
of the millimicrom called by microscopists p/i.
The films were on glass, and the absorption of the glass was
allowed for by control experiments.
It is to be understood that of the whole incident light the pro
portion reflected is first subtracted, and the residue is then called
1 in order to reckon the fraction transmitted of that which enters
the metal, it being understood that the residue which is not trans
mitted (say *68 or *63 in the case of platinum) is absorbed. It
may be that more and better work has been done on the opacity of
metals than this : at any rate there seems to me room for it. I
do not quote these figures with a strong feeling of confidence in their
accuracy. They are to be found in Wied. Ann. vol. xxxv. p. 57.
XXXVI. Note on Mr. Sutherland's Paper on the Cathode Rays.
To the Editors of the Philosophical Magazine.
Gentlemen,
IN the March number of the Philosophical Magazine
Mr. Sutherland considers a theory of the cathode rays
which I published in this Journal in October 1897, and in
which the carriers of the charges were supposed to be the
small corpuscles of which the atoms of the elementary bodies
could, on an extension of Prout's hypothesis, be supposed to
be built up. Mr. Sutherland takes the view that in the
cathode rays we have disembodied electric charges, charges
without matter — electrons — their apparent mass being due to
the energy due to the magnetic force in the field around them
varying as the square of the velocity (see Phil. Mag. xl. p. 229).
I may say that the view that in the cathode rays the con
stancy of the mass arose from the charge being torn away
from the atom, so that we had only the effective mass due to
charge, occurred to me early in my experiments, but except
in the form (which I gather Mr. Sutherland does not adopt),
and which only differs verbally from the view I took, that the
atoms are themselves a collection of electrons, that is, consti
tute an assemblage of particles the individuals of which are
the same as the carriers in the cathode rays ; this conception
seemed to me to be wanting in clearness and precision, and
beset with difficulties from which the other was free. In
the theory which I gather Mr. Sutherland holds of the
cathode rays, we have atoms which are comparatively large
systems ; these can be charged with electricity, of which in
electrons and neutrons we have what correspond to atoms
and molecules, the radius of an electron being about 10~ u cm.
What conception must we form of the connexion between the
416 Prof. J. J. Thomson on the Cathode Rays.
above and the electron when the atom is charged? The charged
atom cannot behave as if the charge were spread over its
surface ; for if it did it would require a potential fall of about
a million volts to separate the electron from the atom. Again,
the value of m/e as determined by the Zeeman effect is of the
samo order as that deduced from the deflexion of the cathode
rays, so that the charge must move independently of the body
charged. The electron thus appears to act as a satellite to
the atom.
A difficulty in the way of supposing that mass is entirely
an electrical effect, and that in the impact of cathode rays we
have electrons striking against much larger masses, is the
large proportion of the energy converted into heat when the
cathode rays strike against a solid. AVhen an electron is
stopped, theory shows that the energy travels off in a pulse of
electromagnetic disturbance, and this energy would only
appear as heat at the place struck if the waves were absorbed
by the target close to the point of impact : if these targets
were made of a substance like aluminium, which is trans
parent to these waves, we should expect much of the energy
to escape in the pulse. As far as I can see the only ad
vantage of the electron view is that it avoids the necessity of
supposing the atoms to be split up : it has the disadvantage
that to explain any property of the cathode rays such as
Lenard's law of absorption, which follows directly from the
other view, hypothesis after hypothesis has to be made : it
supposes that a charge of electricity can exist apart from
matter, of which there is as little direct evidence as of the
divisibility of the atom ; and it leads to the view that cathode
rays can be produced without the interposition of matter at
ail by splitting up neutrons into electrons : it has no ad
vantage over the other view in explaining the penetration of
solids by the rays, this on both views is due to the smallness
of the particles. Until we know something about the vibra
tions of electrons, it does not seem to throw much light on
Eontgen rays to say that these are vibrations of the electrons.
The direct experimental investigation of the chemical
nature (so to speak) of the cathode rays is very difficult, and
though I have for some time past been engaged on experi
ments with this object, they have not so far given any decisive
result.
Yours very sincerely,
J. J. Thomson.
Cavendish Laboratory Cambridge,
March 11th, 1899.
[ 417 ]
XXXVII. Notices respecting New Books.
Harper's Scientific Memoirs. Edited by Dr. J. 8. Ames,
Professor of Physics in Johns Hopkins University, Baltimore. —
I. The Free Expansion of Gases ; Memoirs by GrayLussae, Joule,
and Joule and Thomson. — II. Prismatic and, Diffraction Spectra ;
Memoirs bv J. you Fraunhofer. New York & London : Harper
& Bros., 1898.
r PHESE two volumes form the commencement of a series of
■ memoirs on different branches of physics, each containing the
more important epochmaking papers in connexion with the
subject of the memoir. Professor Ames, in addition to editing the
series, contributes the translations of the papers by GruyLussac
and Fraunhofer in the first and second volume respectively, and
in subsequent volumes such subjects as " The Second Law of
Thermodynamics," " Solutions," " The Laws of Gases," and
" Kontgen Kays " will, among others, receive similar treatment.
Each paper will be enriched by notes aud references, and the
bibliography of the subject will be given in an appendix to each
volume.
The series will serve to bring before Englishspeaking readers
the principal foreign classical papers on physical subjects, and
the reprinting of the papers published in the numerous and
frequently inaccessible journals issued in this country should prove
a great convenience. The list of American physicists who have
undertaken a share of the editing is a guarantee that the work will
be done with the characteristic industry of our friends across the
Atlantic. J. L. H.
XXXVIII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p. 328.]
December 7th (cont.) — W. Whitaker, B.A., F.R.S., President,
in the Chair.
2. 'The Permian Conglomerates of the Lower Severn Basin.'
By W. Wickham King, Esq., F.G.S.
The rocks thus described are the calcareous conglomerates in
cluded in the Middle Permian of the Shropshire type, and exposed
north of the Abberley and Lickey Hills. Three calcareous horizons
occur, interstratified in sandstones or marls and surmounted by
the Permian breccia. It was the opinion of Ramsay and others
that the materials of the calcareous horizons and of the Permian
breccia had been brought from the Welsh border ; but Buckland and
418 Geological Society : —
Jukes, among others, claimed a southern derivation for those of the
Permian breccia, from local hillranges to the south. The latter
view accords with the fact that the pebbles composing these
calcareous horizons, and also the broken fragments constituting the
Permian breccias north of the Abberley and Lickey Hills, are coarser
in the southeasterly direction, and gradually become finer to the
northwest.
The fragments embedded in the Middle Permian calcareous bands
near the Lickey are chiefly of Archaean rocks, but in all the other
districts described there are very few rockfragments older than
Woolhope Limestone. On the other hand, pebbles of dolomitic Wen
lock and Carboniferous Limestones are abundant, while Aymestry
Limestone, Old Bed, Carboniferous, and Lower Permian sandstones
occur in greater or less abundance ; and all these rocks, except the
Carboniferous Limestone, may be seen in situ near at hand to the
south. A summary of work done in the Halesowen CoalMeasure
conglomerates and in the Permian breccia north of the Abberley
and Lickey Hills is given, to bring out one of the lines of argument
adopted.
(1) Ridges near the Lickey were denuded down to the Archaean
rocks in Upper Carboniferous time ; therefore, as might have been
expected, both the adjacent Upper Carboniferous conglomerate and
the Middle Permian calcareous cornstoues are composed of such
fragments of Archaean rocks as are to be found in situ there, or at
Nuneaton ; and the Upper Carboniferous conglomerate is also largely
composed of Palaeozoic rocks identical with those in situ on the
flanks of the Lickey.
(2) The Middle Permian calcareous conglomerates of the other
districts described are for the most part made up of fragments not
older than the Woolhope Limestone, which were presumably derived
by denudation from ridges which had become more extensive.
(3) The Lickey ridges having been denuded to the Archaean rocks
and the more extended area to the Woolhope Limestone, the later
Permian breccias are composed of Archaean fragments near the
Lickey, but of rocks not newer than the Woolhope Limestone in the
other districts north of the Abberley and Lickey Hills.
The author has for several years called the ancient ridges from
which these materials were derived the ' Mercian Highlands,' and
claims that the Palaeozoic and Archaean rocks composing the stumps
of these highlands lie almost entirely buried under the Trias of
the Midlands south and east of the S.E. Shropshire and South
Staffordshire regions.
December 21st.— W. Whitaker, B.A., F.R.S., President,
in the Chair.
The following communications were read : —
1. " On a Megalosauroid Jaw from Rhaetic Beds near Bridgend,
Glamorganshire.' By E. T. Newton, Esq., F.R.S., F.G.S.
On the Torsion Structure of the Dolomites. 419
2. ' The TorsionStructure of the Dolomites.' By Maria M.
Ogilvie, D.Sc. [Mrs. Gordon].
The paper opens with a geueral account of the work of Eichthofen,
Mojsisovics, Bothpletz, Salomon, Brogger, the author, and others
on the Dolomitic area of Southern Tyrol. It then gives the results of a
detailed survey recently made by the author of the complicated strati
graphy of the rocks of the Groden Pass, the Buchenstein Valley, and
the massives of Sella and Sett Sass ; together with the author's inter
pretation of these results, and her application of that interpretation
to the explanation of the Dolomite region in general. The author
concludes that overthrusts and faults of all types are far more common
in the Dolomites than has hitherto been supposed. The arrangement
of these faults is typically a torsionphenomenon, the result of the
superposition of a later upon an earlier strike. This later crust
movement was of Middle Tertiary age, and one with the movement
which gave origin to the wellknown JudicarianAsta phenomena.
The youngest dykes (and also the granitemasses) are of Middle
Tertiary age, while the geographical position of both is the natural
effect of the crusttorsion itself. This crusttorsion also fully
explains the peculiar stratigraphical phenomena in the Dolomite
region, such as the present isolation of the mountainmassives of
dolomitic rock.
The Groden Pass area, first selected for description by the author,
is a distorted anticlinal form running approximately JNT.N.E. and
S.S.W., and including all the formations ranging from th.e Bellerophon
Limestone, through the Alpine Muschelkalk and Buchenstein Beds,
to the top of the Wengen Series. When studied in section, the strata
of the Pass are found to be arranged in a complex fold form, showing
a central anticlinal with lateral wings, limited on opposite sides by
faults and flexures. Strongly marked overthrusting to S.S.E. in
the northern wing is responded to by return overthrusts to N.N.W.
in the southern wing. The strata in the middle limb of the anti
clinal wings bend steeply downwards into kneebend flexures.
Through these run series of normal and reversed faults, into which
has been injected a network of igneous rocks, giving rise to ' shear
andcontact ' breccias, which have previously been grouped as
Buchenstein tuff and agglomerates, and referred to the Triassic
period.
The area of movement of the Groden Pass system is an ellipsoid in
form. Two foci occur within it, where the effects of shear and strain
have culminated. The forces of compression acted not in parallel
lines, but round the area, thus causing torsion of the earthcrust.
Two main faults occur (with a general eastandwest trend) whose
actual lines of direction intersect at a point about midway between
the foci of the torsionellipsoid. These are the chief strike torsion
faults ; many minor ones pass out easterly and westerly from the foci,
forming longitudinal or strike torsionbundles. The strike
system of faults is cut by a series of diagonal or transverse
curved branching faults, with a more or less northeasterly or north
westerly direction. These diagonal faults may cut each other, or
420 Geological Society.
may combine to form characteristic torsioncurves. The author
regards the longitudinal and diagonal faults as constituting one
system. Each portion on one side of the anticlinal form of the
system has its reciprocal on the other side. The Spitz Kofi syncline
on the north is the reciprocal of that of Sella on the south, the
Langkofl on the southwest of that of Sass Songe on the northeast,
and so on.
The anticlinal area of the Buchenstein Yalley is next described.
Here we have a torsionsystem similar to that of the Groden Pass,
and made up of similar elements ; but the western portion of the
anticlinal is much compressed and displaced. Opposing areas of
depression are also found here, that of Sella and Sett Sass on the
north being reciprocated by that of the Marmolata on the south, and
soon. The porphyritesills have here been mainly injected into the
kneebends of the northern wing of the anticlinal form, but igneous
injections and contactphenomena are also met with in some of the
transverse faults.
A full description is given of the sequence and stratigraphy in the
Sella massive — once regarded by some authorities as a Triassic
coralreef. This is an ellipsoidal synclinal area with X.X.E. and
S.S.AV. axes twisted to northeast and southwest. Peripheral over
thrusts have taken place outward from the massive, in such a way
as to buckle up the rocks like a broadtopped fan structure, and these
overthrusts are traced by the author completely round the massive.
A central infold of Jurassic strata occurs on the plateau, where the
Upper Trias has been overthrust inwards on three sides of the infold.
The author next passes in review the results obtained in the area
of Sett Sass, etc. and shows how they all present corresponding
tectonic phenomena.
The district thus studied in detail by the author forms a typical
unit in the structural features of the Dolomite region. It is cut off
to the eastward by the limiting fault (northandsouth N of Sasso
de Stria, and to the westward by the parallel fault of Sella Joch.
These are definite confines, which limit a foursided area, influenced
by the Groden Pass torsionsystem on the north and the Buchenstein
Valley system on the south. The limits of this foursided figure
include a compound area of depression (formed by the Sella and Sett
Sass synclinals) traversed by the diagonal Campolungo buckle.
4 The area displays in a marked degree the phenomena of interference
crossfaults cutting a series of peripheral overthrusts round the
synclines, and parallel flexure faults between the anticlinal buckles
and the synclinal axes/
In conclusion, the author applies her results to the interpretation
of the complexities of the JudicarianAsta region of the Dolomites
in general, aud also to the explanation of the characteristic
structural forms of the Alpine system as a whole.
PHI. Mag. S. 5. Vol. 47. P]. I[[
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XXXIX. The Effects of Temperature and of Circular Mag
netization on Longitudinally Magnetized Iron Wire. By
F. H. Pitcher, M.A.Sc, Demonstrator in Physics, McGill
University, Montreal* .
Objects of the Investigation.
IN commencing these experiments in October 1894, the
original intention was to investigate only the effect of
temperature on the magnetization of iron. With this object
the specimen was heated in a platinum tube, maintained at a
steady temperature by means of an electric current. The
temperature was inferred from the resistance of the platinum
tube, which was very uniformly heated and extended for some
distance beyond the ends of the iron wire. This proved to be
a very perfect method of heating, as the temperature could
be easily varied and accurately regulated and measured. The
current in the platinum tube was also without magnetic effect
on the specimen or the magnetometer, and the specimen was
necessarily at the same mean temperature as the enclosing
tube. A concentric brass tube formed the return lead.
Unfortunately there was some difficulty at the outset in
procuring suitable platinum tubes, and the attempt to make
tubes in the laboratory by rolling up strips of platinum foil
did not prove entirely satisfactory owing to the inferior
quality of the foil. The tubes invariably cracked and became
useless before a complete series of observations had been
* Communicated by Prof. H. L. Calleudar, M.A., F.R.S.
Phil. Mag. S. 5. Vol. 47. No. 288. May 1899. 2 G
422 Mr. F. H. Pitcher on Effects of Temperature and
obtained. The incomplete series of tests obtained in this
manner were, however, of interest as a verification of the
method subsequently adopted.
In the meantime, while awaiting the production of suitable
tubes, it was thought that interesting results might be ob
tained by heating the iron wire with an electric current
passed through the wire itself, and deducing its mean tem
perature from its resistance, with the aid of the formula
verified by Prof. Callendar (Phil. Trans., A. 1887, p. 225) by
the direct comparison of platinum and iron wires. The
objection to this method of heating is that the wire is cir
cularly magnetized by the heating current, and that it is
necessary to disentangle the effects of the temperature change
and of the circular magnetization on the longitudinal mag
netization of the specimen. The effect of the circular field
itself, however, is not without interest*
In order to disentangle these effects, three separate series of
observations were taken for the same range of current or circu
lar field, (1) in a very high vacuum ; (2) in air at atmospheric
pressure; (3) in a current of water. In case (1) a current
of 16 amperes sufficed to heat the wire above its critical
temperature ; in case (2) the highest temperature was
400° C. : in case (3) the heating effect was practically
negligible.
Methods of Measurement Adopted.
The iron wire specimen was magnetized by means of a
specially constructed solenoid, and the intensity of mag
netization I at any time was observed by means of the
deflexion of a magnetometer, the direct effect of the solenoid
being very carefully compensated by means of a balancing
coil in the usual manner. The broadside on position was
adopted for the test in preference to the vertical or the end
on position, as it had been found by preliminary tests that,
if the distance of the specimen from the magnetometer were
suitably adjusted, the broadsideon method agreed much
more closely with ballistic tests of the same specimen than
either of the more usual positions. The value of the Earth's
field H was repeatedly determined by the aid of a Kohlrausch
variometer.
The deflexion of the magnetometer was observed by means
of a telescope and a metrescale of milkglass very accurately
divided. The magnetometer was also provided with suitable
galvanometercoils, so that readings of current and resistance
could be taken on the same scale. The scale of the gal
vanometer was carefullv calibrated throughout, and all the
Circular Magnetization on Magnetized Iron Wire. 423
observations were reduced by means of the correction curve
obtained. The heating and magnetizing currents were
passed through suitable manganin resistances immersed in
oil, so that by observing the deflexion of the galvanometer
when connected successively to the terminals of the manganin
resistances and the specimen itself, the two currents and the
resistance of the specimen could be quickly determined at any
time with an accuracy of at least 1 part in 1000. By varying
the resistance in series with the galvanometer it was possible
to obtain accurately readable deflexions through a very wide
range of current and resistance.
The Magnetizing Solenoid. — The solenoid was wound on a
thin brass tube about 70 centim. long, with an external
diameter of 2*23 centim. The insulation resistance was very
high, special pains having been taken to insulate each layer
with paraffin and paper. The winding was tested for uni
formity by measuring each fifty turns during the process.
The length of the winding was 60  25 centim., containing
4079 turns of No. 24 B & S double silkcovered wire in four
layers, and having a resistance of 28 ohms at 15° G. This
gave a magnetizing field of over 300 C.G.S. with 100 volts
on the terminals. In order to dissipate the heat due to the
magnetizing current at high fields, an internal water circula
tion was provided through an annular space formed by a
second concentric brass tube.
The Mounting for the Iron Wire Specimen. — As the specimen
was enclosed in a vacuumtube, and its resistance at each
temperature determined, a special form of mounting was
necessary.
The ironwire specimen was 0'127 centim. diameter and 26*1
centim. long, or a little over 200 diameters. Its ends were
fused to copper wires *040 centim. diameter and 10 centim.
long ; the diameter of the copper wire being chosen by trial
to give a uniform temperature throughout the whole length
of the iron wire when the heating current was passed through
the circuit.
The ends of these copper wires were tinsoldered and
riveted to stout copper conductors wdiich were brought out
through spiral copper springs to the ends of the containing
tube. The two copper springs, whose function was to take
up the slack of the heated specimen, had each exactly the
same number of turns, and were wound oppositely so that the
direct effect (on the magnetometer) of the current circulating
through them would be compensated. The whole was centered
and kept in place by brass washers which fitted the thin glass
containing tube. The glass tube just fitted the inner brass
2 G2
424 Mr. F. H. Pitcher on Efects of Temperature and
tube, and was made sufficiently long to extend at both ends
beyond the brass tube.
Very fine platinum wires (0*003 centim. diameter) were
attached at 15 centim. apart to the iron wire specimen. They
served as potential leads, and were brought out beyond one
end of the glass containing tube, through sealed capillary
tubes.
The glass tube was made tight at both ends by fitting
brass cups over and filling with fusible alloy, one end of the
tube having been drawn down so that when capped it could
be slipped into the solenoid tube. A copper tube was intro
duced through and soldered in the larger brass cap, to serve
for exhausting. The vacuum was maintained by a fivefall
Sprengel pump, assisted in the early stages by a water pump.
The remaining apparatus consisted mainly of resistance
boxes, rheostats, special arrangements of mercurycup con
tacts, switches, storagebatteries, &c.
Preliminary Tests.
The specimen was of commercial socalled soft iron wire,
and was carefully annealed and polished before mounting.
The vacuumtube containing it was connected to the Sprengel
pump and a high vacuum maintained while the wire was
being heated by the current. It was observed by the eye
that the heating was very uniform, the whole becoming
an even red right up to the ends, at a high vacuum. The
zeropoint or resistance at 0° C. of the specimen was now
obtained. It was then placed in the solenoid and the
equivalent magnetic length determined. This was found to
be a little over 20 centim., and the magnetometer distance
was arranged so that slight changes of the length had a
minimum effect on the magnetometer readings. A pre
liminary test for the magnetic quality of the iron at ordinary
temperatures w T as first made. It was found before further
annealing to be fairly hard, having a hysteresis loss for
B = 17,000, of 16,000 ergs per cub. centim., and a permeability
at that induction of 500.
After annealing several times in a vacuum, the loss at
nearly the same induction had fallen to 6000 ergs, and finally,
after successive annealings, arrived at the extraordinarily low
value of 557 ergs for B = 3500 at ordinary temperatures.
This, in spite of the fact that the specimen was only com
mercial wire, is almost as good as the best specimen of
transformer iron tested by Ewing. By this time the wire
had settled down to a very steady magnetic state, as shown
Circular Magnetization on Magnetized Iron Wire. 425
by successive tests before and after heating. Before pro
ceeding further the zeropoint of the specimen was again
tested, and was found to agree to within 1/10 of one percent,
with the previous determination. There was no trace of
oxidation.
The Observations.
The method of taking the observations was as follows : —
First, the containing tube was exhausted. The magnetometer
deflexions were observed at longitudinal fields ranging from
1 to 30, with currents in the wire varying from to 1 6 amp.
It was previously observed that the wire was practically
demagnetized at 16 amps., which corresponds to a temperature
of 750° C.
The current in the solenoid was reversed several times
before each reading of the magnetometer deflexion, thus
ensuring a reversal curve. The current in the wire was kept
constant for each reversal curve. Its value with that of the
corresponding resistance of the wire was observed at intervals
along the curve. The effect of residual thermal currents in
the heating circuit was eliminated by reversal of the current
in the wire.
The Sprengel pump was kept running during the whole
set of observations in order that the gases given off from
the heated iron and copper, as well as air which might leak
in owing to imperfect sealing, might not affect the vacuum.
Under these conditions the vacuum was kept very high and
constant, and the iron wire remained bright throughout the
whole series of tests.
On the completion of this set of observations the vacuum
was let down, the containing tube disconnected from the
pump, and a similar set taken in air. All the conditions
remained the same as before, except that the wire was tested
in air instead of in a high vacuum, and was therefore at a
necessarily lower temperature for the same heating current.
As soon as possible afterwards, two similar series of obser
vations were taken at much higher fields, varying from 50 to
300. The conditions were exactly the same in this case as in
the lower fields, except that the controlling field of the
magnetometer had to be strengthened, and that the vacuum
at which the higher temperature observations were taken was
slightly less perfect.
At this stage the zeropoint of the specimen was again
tested and was found to agree with the two previous deter
minations, within the limits of accuracy of the method. It is
interesting to observe that the electrical resistance was
426 Mr. F. H. Pitcher on Effects of Temperature and
practically unaffected by magnetization, and that the reduc
tion of the hysteresis loss to onethird of its original value
was unaccompanied by any measurable change of conductivity.
Specimen tables of the reduced observations in vacuum and
in air are here exhibited.
Table I. — Longitudinal Fields 130.
In Vacuum.
In Air.
Current in Wire 1245 amp.
Eesistance of Wire 008324 w.
Temperature of Wire 552° 0.
Current in Wire 12*60 amp.
Eesistance of Wire 003447 w.
Temperature of Wire 224° C.
I.
H.
I.
H.
434
673
861
1084
1350
1735
2861
4376
9630
099
145
178
224
275
360
627
1010
3030
208
427
588
820
1080
1470
2690
4325
10030
099
145
175
223
275
359
627
1008
3025
Table II.— Longitudinal Fields 50290.
In Vacuum.
In Air.
Current in Wire 12 amp.
Eesistance of Wire 007312 w.
Temperature of Wire 496° C.
Current in Wire 1T60 amp.
Eesistance of Wire 03035 w.
Temperature of Wire 187° C.
I.
H.
I.
H.
1235
1359
1397
1437
512
939
1679
2904
1254
1410
1528
1633
512
939
1670
2903
Circular Magnetization on Magnetized Iron Wire. 427
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Figs. 1 and 2. Containing Results of Direct Observation.
428 Mr. F. H. Pitcher on Effects of Temperature and
The Curves of fig. 1. — The magnetic observations were
taken at thirteen different fields, curves for seven of which
have been plotted in fig. 1. The longitudinal field of the
solenoid is indicated for each pair of curves. Each pair is
drawn for the same field. Abscissae represent current in
the wire and ordinates longitudinal intensity of magnetization.
The curves drawn in full lines are for the observations taken
in a vacuum and at a higher temperature than those in dotted
lines, which are for the observations taken in air. Where the
curves coincide, showing no effect of temperature, full lines
are drawn.
The curves shown in fig. 2 are the corresponding tempe
raturecurves plotted to current in the wire. The lower of
these curves is that for the specimen in air. The points
marked with a cross belong to the observations in fields above
30 c.G.s. The middle curve is the temperaturecurve in
vacuum for the higher fields ; while the top curve is that for
lower fields in vacuum, from 30 down. The temperature
difference between the dotted and fullline curves for any
current in the wire can at once be found by consulting the
corresponding temperaturecurves in fig. 2 on the same
ordinate.
Considering the tables and the curves of figs. 1 and 2, it
will be noticed that the known behaviour of soft iron at constant
fields, as temperature advances, is well displayed.
In high fields the dotted and fullline curves of each pair
separate almost from the start, and do not meet in any part
of their course, showing a continual decrease in intensity of
magnetization from the beginning as temperature increases.
At a field of 10 c.G.s. there appears to be no change in
the intensity of magnetization for a temperaturedifference of
350° C, as shown by the fourth curve (fig. 1) from the top,
together with the top and bottom curves (fig. 2) .
At still lower fields the intensity begins to increase for a
comparatively small temperaturedifference, as shown by the
two lower pairs of curves, fig. 1. The effect of the circular
field on the longitudinal component is here very marked. If
the dotted curves, where the temperature is less in evidence,
be considered, it will be seen that, in high longitudinal fields,
as the circular field increases there is but little change in the
longitudinal intensity. Somewhere between a longitudinal
field of 30 and 10 a point of inflection occurs, and the curves
below are changed in form entirely. At fairly low fields
(from 3*5 downwards) the drop with small increments of
circular field is at first very great, but soon reaches a limit ;
and the curves become very flat.
The explanation of these effects is that for high longitudinal
Circular Magnetization on Magnetized Iron Wire. 429
fields the permeability is very small. Therefore the circular
field would have at first only a slight effect in diminishing the
longitudinal intensity. On the other hand, in lower fields
the permeability is many times greater, and hence the effect
of the circular field is much more marked, until the direction
of the resultant field swings around nearer to the direction of
the circular field, when the rate of change in the longitudinal
intensity becomes very slow.
Temperature Curves, fig. 2. — The curious wave which occurs
at the upper end of each hightemperature curve {in vacuo)
may be partly due to the sudden change in the temperature
coefficient of iron at high temperatures *, and partly also, in this
particular case, to the effect on the vacuum of gas given off
from the wire. The pump may not have been able to exhaust
at a sufficiently high rate.
Hopkinson (Phil. Trans, vol. clxxx.) investigated the resist
ancetemperature curves of soft iron and steel at high tempe
ratures up to 900° C. The temperature was inferred from
the resistance of a copper wire enclosed with his specimens,
apparently on the assumption of a constant temperature
coefficient for copper. He found a sudden drop in the tem
peraturecoefficient for soft iron and steel between 800° and
900° C. beyond the critical point. It seems desirable that
this should be tested up to higher temperatures by comparison
with a platinum pyrometer.
Method of Distinguishing the Effects of Temperature and
of Circular Magnetization.
By treating the ordinates of the curves in fig. 1 as one
component of the resultant intensity the temperaturevariation
of the magnetization of iron at high fields can be worked out
to a fairly accurate result.
The first step in the reduction was to obtain a family of
curves {a) of average resultant I and H at different tem
peratures. These were compared with a similar set (b) from
which temperatureeffect had been eliminated. Then by
treating the drop between corresponding curves of the first
and second set — at the same resultant fields — as due to tem
perature, the temperatureeffect on the resultant intensity
was obtained.
The average circular field in the wire was taken equal to
two thirds of the field at the periphery. This was compounded
with the longitudinal field to give the average resultant field
due to the two magnetizing forces. The longitudinal field
previous to compounding was corrected for the effect of the
* A rapid increase of a similar character was observed by Callendar to
occur just below the critical point.
430 Mr. F. H. Pitcher on Effects of Temperature and
ends of the specimen. The corresponding value of the average
resultant permeability was taken from the fullline curves
(fig i).
Effect of Circular Magnetization (fig. 3).
Before obtaining the second set (b) of average resultant I
and H curves with which the above were compared, it was
necessary to eliminate the effect of temperature. This was
done by taking the drop between any two points on the dotted
and fullline curves (fig. 1) which are at the same tempe
rature as due to circular magnetization, an assumption which
is very nearly correct, especially in the higher fields.
In this way the family of curves in fig. 3 was obtained.
/sect
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Effect of Circular Field at Constant Temperature 18° C. on the
IH Curves of Longitudinal Magnetization.
The ordinates are longitudinal magnetization,, and the abscissae
longitudinal field. The curves in the lower part of the figure
are the continuation in higher fields of those above. They
are plotted to the same scale of I, but the H scale is reduced
Circular Magnetization on Magnetized Iron Wire. 431
ten times. The curves are all drawn for the same tempe
rature, viz. 18° C, and the current in the wire is indicated in
each case. The correctionline for the length of the specimen
is also drawn.
Method of Deducing the Temperature Curves (fig. 4).
By compounding the ordinates of the curves in fig. 3 with
the corresponding value of the average circular intensity, the
set of resultant I and H curves (b) at constant temperature
for comparison with the corresponding set (a) at different
temperatures was obtained. These two sets of curves are not
shown in the figures. They were only a step in the reduction,
and were not intrinsically interesting.
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Temp, /n Degrees Cent".
Effect of Temperature on the Magnetization of Iron in High Fields.
The final result showing the effect of temperature in high
fields is shown in fig. 4. Here ordinates represent average
resultant intensity and abscissa? temperature in degrees centi
grade. Each curve is drawn for a constant average resultant
field.
432 Effects of Temperature on Magnetized Iron Wire.
Verification and Discussion of Results.
It is interesting to notice in fig. 3 the limiting effect of the
circular magnetization in high fields. The curves up to
1 2 amps, in the wire almost coincide at a. longitudinal field
of 150 C.G.S.
To test how far the results in fig. 3 were reliable, the wire
was mounted in a glass tube and a water circulation allowed
to flow through the tube in contact with the wire. Thus the
temperature of the wire was kept practically constant for all
values of the current. The results of these tests agreed very
closely indeed with those shown in fig. 3, even down to a field
of 10 C.G.S.
Klemencic (Wied Ann. vol. lvi. p. 574) investigated the
circular magnetization of iron wires together with the axial
magnetization by a different method. By including the wire
as an arm of a Wheatstone's bridge and using a ballistic
galvanometer, the circular magnetization was deduced from
the observed value of the selfinduction for different currents
in the wire. Here the change of temperature of wire intro
duces difficulties. It seems that the magnetometer method,
when the wire is kept at a constant temperature, is much
simpler and less troublesome.
The results shown in fig. 4 were found to agree closely with
the tests obtained for soft iron by the platinumtube method.
The point of demagnetization was obtained a trifle lower by
the latter method, and the initial slope of the curve at a field
of 290 c.G s. was a little less. It will be noted that the point
of demagnetization is not absolutely sharp ; the curves suddenly
change their direction and I decreases more slowly. This
was also investigated in special tests made by the platinum
tube method. The value of I at 750° C. was observed to be
about 7 C.G.S.
These results are found to agree very well with those for
soft iron in high fields obtained by Curie (Comptes Rendus,
vol. cxviii. p. 859). He heated his specimen in a platinum
heatingcoil and measured the temperature inside the coil by
a thermocouple. The point of demagnetization which he
obtained is rather higher than that obtained by the platinum
tube or currentinthewire methods, but not so high as that
given by Hopkinson (loc. cit.) who used a copper wire for
his temperature measurements. More recently a paper has
appeared by Morris (Phil. Mag. vol. xliv. Sept. 1897), who
employed the same method of heating as Curie, but measured
the temperature with a platinum wire. It will be seen that
the method of heating with a coil is less perfect than with
Resistance and Inductance of a Wire to a Discharge. 433
a platinum tube, and less simple than the current in the
wire.
The results of the foregoing experiments were communicated
to Section Gof the British Association at Toronto, and a brief
abstract appears in the B. A. Report for 1897, but the curves
were not reproduced.
In conclusion, I should like to thank Professor Oallendar
for kind suggestions and other assistance.
Macdonald Physics Laboratory,
December 20, 1898.
XL. The Equivalent Resistance and Inductance of a Wire
to an Oscillatory Discharge. By Edwin H. Bakton,
D.Sc, F.R.S.E., Senior Lecturer in Physics, University
College, Nottingham *.
I.N an article in the Philosophical Magazine for May
1886 f, Lord Rayleigh, whilst greatly extending Max
well's treatment of the selfinduction of cylindrical con
ductors, confined the discussion of alternating currents to
those which followed the harmonic law with constant ampli
tude. The object of the present note is to slightly modify
the analysis so as to include also the decaying periodic
currents obtained in discharging a condenser and the case of
the damped trains of highfrequency waves generated by a
Hertzian oscillator and now so often dealt with experi
mentally. In fact, it was while recently working with the
latter that the necessity of attacking this problem occurred
to me.
Resume of previous Theories. — To make this paper in
telligible without repeated references to both Maxwell and
Rayleigh, it may be well to explain again the notation used
and sketch the line of argument followed.
The conducting wire is supposed to be a straight cylinder
of radius a, the return wire being at a considerable distance.
The vector potential, H, the density of the current, w, and
the " electromotive force at any point " may thus be con
sidered as functions of two variables only, viz., the time, t,
and the distance, r, from the axis of the wire. The total
current, C, through the section of the wire, and the total
electromotive force, E, acting round the circuit, are the
variables whose relation is to be found. It is assumed that
H=S + T + V 2 + .... +T,r* . . . (1)
* Communicated by the Physical Society : read January 27, 1899.
f " On the Selfinduction and Resistance of Straight Conductors."
434 Dr. E. H. Barton on the Equivalent Resistance and
where S, T , T l5 &c, are functions of the time. A relation
between the T's is next established so that the subscripts are
replaced by coefficients. The value of H at the surface of the
wire is equated to AC, where A is a constant. This leads to
Maxwell's equation (13) of art. 690. The magnetic permea
bility, //,, of the wire, which Maxwell had treated as unity, is
now introduced by Lord Rayleigh, who thus obtains in place
of Maxwell's (14) and (15) the following equations : —
^=^* + i^^ + ;; + ii^^a? + >' (2)
ACS = J + ^ + g:?;g +.... + lit2 , t<n ,^+....j > (o)
where a, equal to //R, represents the conductivity (for steady
currents) of unit length of the wire.
By writing
»(«) = ! + «+ pr^+ — + 1 2 [ / tiW 2 +  • W
equations (2) and (3) are then transformed as follows
ctt ~~ dt
d$ A dC* ,( d\ dT
ty^dtjdt' • • • w
C = ^).f: .... (6)
we have further
5^ (7)
I dt [n
Lord Rayleigh then applies equations (5), (6), and (7) to
sustained periodic currents following the harmonic law, where
all the functions are proportional to e ipt , and obtains
E=R'C+^L'C, (8)
R / and 1/ denoting the effective resistance and inductance
respectively to the currents in question. The values of R'
and \J are expressed in the form of infinite series. For high
frequencies, however, they are put also in a finite form,
since, when p is very great, equation (4) reduces analytically
to
*W=F7 = V' (9)
£ V7T •*•
* AC is printed here in Fhil. Mag., May 1886, p. 387 ; but appears to
be a slip for A — .
v dt
Inductance of a Wire to an Oscillatory Discharge. 435
so that
#{X)
= xh (10)
Equivalent Resistance and Inductance for Oscillatory Dis
charges. — To effect the object of this paper we must now
apply equations (5), (6), and (7) to the case of logarithmically
damped alternating currents where all the functions are
proportional to e^~ k)pi .
The value of E so obtained must then be separated into
real and imaginary parts as in (8), and then, together with
the imaginary quantities, must be collected a proportionate
part of the real ones so as to exhibit the result in the form
E = B"C+(e%L // C (11)
The quantities denoted by B/' and \J' in this equation will
then represent what may be called the equivalent resistance
and inductance of length / of the wire to the damped periodic
currents under discussion. For, the operand being now
e dk)pt^ jfa Q time differentiator produces (i — k)p, and not ip
simply as in equation (8) for the sustained harmonic currents.
dT
Thus (5), (6), and (7), on elimination of 8, Z, and jj, give
E _/;_ k \ n „\  <Kip*pkp*fi) (m
Now we have
thus
^) 1+ 2l2 + 48180 + *" * {l6)
cj)(ipufJL — /cpa/jb)
<fi (ipufjL — kjjafA)
+i[ip^+ ^v 1 ^p z «V  m ]^py w } . (u)
Hence, substituting (14) in (12) and collecting the terms
as in (ll), we find that
R" 1 + i 2 s A(l + A«) 3 , 3 l2F3/t 4 4 4
TT =1 + T2AV+ — jjW fgo AV • ••, (15)
= l+ToP 2a Vf«nAV...., . . (17)
436 Dr. E. H. Barton on the Equivalent Resistance and
and
or J (16)
L«=z[A+Mi+ l^^^^V^j^^V..)] J
Putting £ = in these equations and denoting by single
dashed letters the corresponding values of the resistance and
inductance, we have
t*' 1 i ■*■ 2 2 2
s =i+_^v lTo
and
L'=Z[A + Ki^*V...)], • • (18)
which are Lord Rayleigh's wellknown formulae * for periodic
currents of constant amplitude.
By taking the differences of the resistances and inductances
with damping and without, we have at once
*L^K =k yaY+ ^^V«V+ (19)
and
L''L=^(j.^+V"V ••••)• • • (20)
These show that if the frequency is such that a few terms
sufficiently represent the value of the series, then both resist
ance and inductance are increased by the damping.
High Frequency Discharges. — Passing now to cases where
p is very great, as in the wavetrains in or induced by a
Hertzian primary oscillator, we have from equation (10),
where s= y/l + k 2 and cot6 = k.
On substituting this value of <p/4>' in equation (12) and
collecting as before, we obtain the solution sought, viz.:
^q = (*H*ps d )* cos 2+(i—k)pya A + *s/a/j,s/p cos A; (22)
whence R" , ,. x 6 nON
g = (a/ips z ^cos^, ^23)
Equations (19) and (20), p. 387, loc. cit.
Inductance of a Wire to an Oscillatory Discharge. 437
and L" . , , , y
r^ =aA+ (<zfLS/p)» COS^
MHSWf ' ' ' <21>
Discussion of the Results for High Frequencies.
On putting & = 0, in equations (23) and (24), to reduce to
7T
the case of sustained simple harmonic waves, 5=1, 0=\
u
whence, denoting by single dashes these special values of R"
and L", we obtain
g = Sfatp ; (25)
"'{*+</£} > • • • • <*>
which are Lord Rayleigh's highfrequency formulae*.
Referring again to equations (23) and (24), we see that
for a given value of _p, if k varies from to co , the factor
involving s increases without limit while that involving
increases to unity. Hence, with increasing damping, it
appears that R" and U f each increase also, while ever the
equations remain applicable. Now an infinite value of k
involves zero frequency f. And a certain large, though
finite, value of k would prevent the frequency being classed as
u high."
Dividing equation (23) by (25) gives
g'=(2 s 3 )icos = K S a y (27)
Thus, for a given value of k, the ratio W/Rf is independent
of the frequency of the waves. It is therefore convenient to
deal with K a function of k only, rather than with R"/R
which is a function of p also.
Differentiating to k, we have
1Tr ok cos 77 + sin r
1k~ 711 ' (28)
* Equations (26) and (27), p. 390, loc. cit.
t This follows from the fact that electric currents or waves generated
by an oscillatory discharge may be represented by e~ kpt cos pt, in which
Jcp is finite, so k is infinite only when p is zero.
Phil. Mag. S. 5. Vol. 47. No. 288. May 1899. 2 H
438 Dr. E. H. Barton on the Equivalent Resistance and
Fig. 1. — Exhibiting graphically K = R"/R' as a function of
k, the damping factor.
flUH ■■
Inductance of a Wire to an Oscillatory Discharge. 439
which is positive for all values of k from to go , hence K
increases continuously with k. For & = 0, this becomes
'dK\
. dk A =0 ~~
7T
sin
4
~V2
which assists in plotting K as a function of k.
Differentiating again, we obtain
dk 2
, (7 + 3*«)cosg + 2*sin 5 }
6
(29)
(30)
Since this expression is positive for all values of k from
to oo , we see that K plotted as a function of k is a curve
which is always convex to the axis of k. Thus the nature of
R'yR' as a function of k is sufficiently determined.
Pairs of corresponding values of K and k for a few typical
cases are shown in the accompanying Table, and part of the
curve coordinating them is given in fig. 1. It is not necessary
to plot much of the curve, as only a small part of it can
apply to any actual case. For, although k may have any
positive value up to go , the high values of k, as already men
tioned, correspond to low values of p and so exclude them
from the application of the highfrequency formula.
Table showing the values of K = R"/B/, the ratio of equiva
lent resistances to waves with damping and without.
Damping Factor,
h = cot 0.
Subsidiary quantities involved.
Ratio of
Resistances
K=^R fR .
0/2.
6  2 = lfP.
45°
1
1
i =00798
4tt
42° 44'
1006362
T044 nearly
J* =00955
107T
42° 16'
100913
1054 „
i =01595
2ir
40° 28'
102614
1097 „
— =0319
7T
36° 9'
11018
1228 „
1
22° 30'
2
2197 „
2
13° 17'
5
4 602 „
3
9° 13'
10
785
2 H 2
440 Equivalent Resistance of a Wire to Oscillatory Discharge*
Fig. 2. — Instantaneous Form of Wave train for fc = l, whence
R"/R / = 2197.
Figure 2 shows the form of a wavetrain for which k = l
and K=2*197. That is to say, in this extreme case where
all the functions vary as e^' 1 ^, and the wavetrain passing
a given point of the wire is accordingly represented by
e^cospt, then the equivalent resistance is 2*197 times that
which would obtain for simple harmonic waves uniformly
sustained and of the same frequency.
Figure 3 represents the form of the wavetrains generated
and used in some recent experiments on attenuation*. In
this case the value of k was approximately vtc— , or the
logarithmic decrement per wave =27rk = m 6, and the corre
sponding value of K, the ratio of Bf'/R', is T054. Now in
* "Attenuation of Electric Waves along a Line of Negligible Leakage,"
Phil. Mag. Sept. 1898, pp. 296805.
Diffraction Fringes and Micrometric Observations. 441
Fig. 3. — Instantaneous Form of WaveTrain for &='3/7r, whence
R'VR' = ] 0.54.
wmmm
the experiments just referred to the frequency was '65 X 10°
per second, and R'/R became 31*6. Hence R"/R has the
value 31*6 x 1054=333 nearly. Thus, writing e  w ^ 2Lv for
the attenuator of the waves along the wires instead of
4f g EV/2L» " ncreases the index by about five and a half per cent.,
and so brings it by that amount nearer to the value deter
mined experimentally.
Univ. Coll., Nottingham,
Nov. 29, 1898.
XLI. On certain .Diffraction Fringes as applied to Micro
metric Observations. By L. N. Gr. Filon, M.A., Demon
strator in Applied Mathematics and Fellow of University
College, Londonf.
1. rTVEE following paper is largely criticism and exten
X sion of Mr. A. A. Michelson's memoir "On the
Application of Interference Methods to Astronomical Obser
vations," published in the Phil. Mag. vol. xxx. p. 256, March
1891.
* See Equation (2) p. 301, Phil. Mag. Sept. 1898.
t Communicated by the Physical Society : read November 2o, 1898.
442 Mr. L. N. G. Filon on certain Diffraction Fringes
Light from a distant source is allowed to pass through two
thin parallel slits. The rays are then focussed on a screen
(or the retina of the eye) and interferencefringes are seen.
If the distant source be really double, or extended, the fringes
will disappear for certain values of the distance between the
slits. This distance depends on the angle subtended by the
two components of the double source or the diameter of the
extended source.
Mr. Michelson, however, in obtaining his results treated
the breadth of the slits as small compared with the wave
length of light and their length as infinite. This seems un
justifiable a priori. The present investigation takes the
dimensions of the slits into account.
2. Suppose we have an aperture or diaphragm of any
shape in a screen placed just in front of the objectglass of a
telescope (fig. 1).
Fig. 1.
s* — 
, / ,
< *:
£^^1,
P
A'
/ J^'
1
A
Let the axis of the telescope be the axis of z. Let the
axes of oc and y be taken in the plane of the diaphragm OQ
perpendicular to and in the plane of the paper respectively.
Let S be a source of light whose coordinates are U, V, W.
Let Q be any point in the diaphragm whose coordinates are
(#, y). Let A P be a screen perpendicular to the axis of the
telescope, and let (p, q) be the coordinates of any point P on
this screen. Let A'P' be the conjugate image of the screen
AP in the objectglass.
Let b = distance of centre C of lens from screen AP.
b'= „ „ „ „ image of screen AP.
/= distance of diaphragm OQ from plane A'P'.
Then if, as is usual, we break up a wave of light coming
as applied to Micrometric Observations. 443
from S at the diaphragm, the secondary wave due to the
disturbance at Q would have to travel along a path QRTP in
order to reach a point P on the screen, being regularly
refracted.
But since P f is the geometrical image of P, all rays which
converge to P (i. e. pass through P) after refraction, must
have passed through P' before refraction, to the order of our
approximation.
Hence the ray through Q which is to reach P must be
P'Q.
Moreover, P and P' being conjugate images the change of
phase of a wave travelling from P' to P is constant to the first
approximation and independent of the position of Q.
Now the disturbance at P due to an element dx dy of the
diaphragm at Q is of the form
^^ sin 2^ SQQRm. BTTP),
where X is the wavelength, t is the period, A is a constant,
(i is the index of refraction of the material of the lens, and h
is put instead of QP outside the trigonometrical term, because
the distance of the lens from the diaphragm and the inclination
of the rays are supposed small.
But P'Q + QR + fiUT + TP = constant for P.
Therefore the disturbance
But
= A^ sin ^^ SQ + P'Q const.)
SQ 2 =(.zU) 2 + G/V)*rW 3
P # Q , =(*+^) f +(y+^) i +/ s 
Now in practice x, y, p, q are small compared with b\ b, f,
or W ; U and V are small compared with W. Neglecting
terms of order IP/W 3 , xW/W, & c ., we find
In like manner
FQZ/a ft— 4 ^ + ? 2 4^4 W 4 L f!±ff!
remembering that / is very nearly equal to V because the
diaphragm is very close to the lens.
441: Mr. L. N. G. Filon on certain Diffraction Fringes
Hence the difference of retardation measured by length in
air
P'QSQ=const. +( + ^)» + (f + \)y
+*(*+/>(? If)
If now the geometrical image of S lie on the screen AP
{i.e. if the screen is in correct focus) b' = W and the last
term disappears.
If, however, the screen be out of focus \jV is not equal to
1/W, and the term in x^+y 2 may be comparable with the
two others, if b' be not very great compared with b. Thus
we see that appearances out of focus will introduce expres
sions of the same kind as those which occur when no lenses
are used.
We will, however, only consider the case where the screen
is in focus. Let — u and — v be the coordinates of the geome
trical image of S ; then
u/b = TJ/W, v/b=Y/W.
The difference of retardation measured by length in air is
therefore of the form
Hence the total disturbance at P (integrating over the two
slits) is given by the expression
a+k h
a k —h
a + k h
C j Cj ' 27r / xt . P + n . 9 + v \
A
■a — k
where A = a constant,
2a = distance between centres of slits,
2 k = breadth of either slit,
21i — length of either slit.
This, being integrated out, gives
^ 2ALX . 2irt 27rq+v . 2irq+v, . Zirp + u,
D=— 57 — i — \7 — r~\ sm — cos— ^^— asm— ^— Asm r ~* h,
ir 2 (p\u)(q + v) r X b X b X b
as applied to Micrometric Observations. 445
whence the intensity of light
4A 2 i 2 A* 2™ , . . 9 2*k, N . ,2tA,
I = ~T7 — , \2? — i — ^2 cos "FT W + w ) Sln T^  (? + v ) sm ~TT" (P
This may be written
64AW 2™ , x Sm Tx ^ v) Bm !x ( ^ + " }
This gives fringes parallel to x and y : k being very small
compared with a, the quick variation term in v is
2 27ra , .
° TxT ^ + ^
Consider the other two factors, namely :
sin^7— (g f r) sm 2 ^— (p + u)
and
sin <27
If we draw the curve y= — %~ (see fig. 2), we see that
OS
these factors are only sensible, and therefore their product is
only sensible, for values of p + u and q + v which are numeri
cally less than bX/2h and bX/2k respectively.
Fig. 2.
Hence the intensity becomes very small outside a rectangle
whose centre is the geometrical image and whose vertical
and horizontal sides are bX/k and bX/h respectively.
This rectangle I shall refer to as the " visible " rectangle of
the source.
Inside this rectangle are a number of fringes, the dark
lines being given by
q f v = — . bX
* 4«
and the bright ones by q+ v = nbX/2a.
446 Mr. L. N. G. Filon on certain Diffraction Fringes
The successive maximum and minimum intensities do not
vary with a. Hence, what Mr. Michelson calls the measure
of visibility of the fringes, namely the quantity
I, Is
Il+I*'
where I l5 1 2 are successive maximum and minimum intensities,
does not vary with the distance between the slits. The only
effect of varying the latter is to make the fringes close up or
open out. Hence for a pointsource of light the fringes
cannot be made to practically disappear.
3. Consider now two pointsources of light whose geo
metrical images are J 1? J 2 , and draw their
(fig. 3).
FIr. 3.
visible rectangles
: J
Jj
To get the resultant intensity we have to add the in
tensities at every point due to each source separately.
Then it may be easily seen that the following are the
phenomena observed in the three cases shown in fig. 3: —
(1) The two sets of fringes distinct. Consequently no
motion of the slits can destroy the fringes. In this case,
however, the eye can at once distinguish between the two
sources and Michelson's method is unnecessary.
(2) Partial superposition : the greatest effect is round the
point K, where the intensities due to the two sources are
very nearly equal. If v' — v be the distance between Jj and
J 2 measured perpendicularly to the slits, so that (v' — v)/b is
the difference of altitude of the two stars when the slits are
horizontal, then over the common area the fringe system is
(a) intensified if v' —v be an even multiple of bX/Aa, (b)
weakened, or even destroyed, if v' — v be an odd multiple of
b\/4:a. For in case (a) the maxima of one system are super
posed upon the maxima of the other, while in case (b) the
maxima of the one are superposed upon the minima of the
other. This common area, however, will contain only com
paratively faint fringes, the more distinct ones round the
as applied to Micro metric Observations. 447
centres remaining unaffected. We may suppose case (2) to
occur whenever the centre of either rectangle lies outside the
other, i. e. whenever v , — v>b\/2k, u / — u>b\/2h,u' — u being
the horizontal distance between J l and J 2 .
(3) Almost complete superposition of the visible rectangles.
The fringes of high intensity are now affected. These are
destroyed or weakened whenever a is an odd multiple of
b\/4:(v f — v), provided that the intensity of one source be not
small compared with that of the other.
Case (3) may be taken to occur when v' — v <b\/2k and
u' ' — u< b\/2h.
The smallest value of a for which the fringes disappear is
b\/i(v f v).
If v r —v be very small, this may give a large value of a.
Now a double star ceases to be resolved by a telescope of
aperture 2r if (v' — v)/b<\/2r, and when this relation holds
the smallest value of a for which the fringes disappear is not
less than r/2, which is the greatest separation of the slits
which can conveniently be used. Hence the method ceases
to be available precisely at the moment when it is most
needed.
(4) Mr. Michelson, in the paper quoted above, noticed this
difficulty, and described an apparatus by means of which the
effective aperture of the telescope could be indefinitely
increased. He has not shown, however, that the expression
for the disturbance remains of the same form, to the order of
approximation taken, and he has made no attempt to work
out the results when the slit is taken of finite width, as it
should be.
In his paper Mr. Michelson describes two kinds of appa
ratus. I shall confine my attention to the second one, as
being somewhat more symmetrical.
So far as I can gather from Mr. Michelson's description,
the instrument consists primarily of a system of three mirrors
a, b, c and two strips of glass e, d (fig. 4). The mirrors a and
b are parallel, and c, d, e are parallel. Light from a point P
in one slit is reflected at Q and R by the mirrors a and b, is
refracted through the strip e, and finally emerges parallel to
its original direction as T U. Light from a point P' in the
other slit is refracted through the strip d, and reflected at S'
and T' by the strips e and c.
I may notice in passing that the strip e should be half
silvered, but not at the back, for if the ray S' T is allowed to
penetrate inside the strip and emerge after two refractions
and one reflexion, not only is a change of phase introduced,
owing to the path in the glass, which complicates the analysis,
448 Mr. L. N. Gr. Filon on certain Diffraction Fringes
but tlie conditions of reflexion, which should be the same for
all four mirrors, are altered, and this changes the intensities
of the two streams. We shall see afterwards that this silver
ing can be done without impeding the passage of the trans
mitted stream, as it will turn out thnt the two streams must
be kept separate.
p
Fig. 4.
__n\q_. — W
mC
A %l — 'Vm¥\
!__— —  — — ~~"
\
c i^
Suppose then that a plane wave of light whose front is
M M / is incident upon the diaphragm. Let us break the wave
up, as is usual, in the plane of the diaphragm. Let Z be a
point on the screen whose cordinates are (p, q) at which the
intensity of light is required.
Then if C be the centre of the objectglass, the direction in
which rays T U, T' U' must proceed in order to converge to
Z after refraction is parallel to C Z.
Hence PQ, RS, TU, P'Q', R'S', T'U' are ail parallel to OZ,
and the directioncosines of CZ are
*V + ? 2 + 6 2 ' s/'pi + q^ + tf' ^zfi + tf + o*
I shall assume that the strips e and d are cut from the same
plate and are of equal thickness. This will sensibly simplify
the analysis, though, as I think, it would not materially influ
ence the appearances if the strips were unequal.
If, however, we suppose them equal, we may neglect the
presence of strips, as far as refraction is concerned, since
clearly the retardation introduced is the same for all parallel
rays.
If now UU f be a plane perpendicular to CZ, then, since we
know that rays parallel to TU, T'U' converge to a focus at Z,
the only parts of the paths of the rays which can introduce a
as applied to Micrometric Observations. 449
difference of phase are
MP + PQ + QR + RS + TU for one stream,
M'P' + PQ' + R'S' + S'T' + T'U' for the other stream.
Produce PQ, P'Q' to meet it in V and V and let N, N' be
the feet of the perpendiculars from R and S' on PQ, T'U'
respectively.
Thus we may take the change of phase as due to the
retardation
(MP + PV)+(NQ + QR)
for diffraction at one slit, and to the retardation
(M'P + P V) + (ST + T'N')
for rays proceeding from the other slit.
The terms in the first brackets give us the expression which
we had before, viz. : —
As to the other terms
NQ + QR = QR (l + cos26)==«^i^ r ?^=2acosW),
v T COS (p
where a is the distance between the mirrors a, b and <£> is the
angle of incidence of any ray on these mirrors.
Similarly S'F + T'N' = 2/3 cos i/r, where /3 is the distance
between e and c and ty the angle of incidence of any ray
upon e and c.
Now if the mirrors a and b are inclined to the plane of the
diaphragm at an angle 0, c, d, e at an angle ( — &), then
, q sin + b cos
cos <p —
cos yjr
— q sin & + b cos 0'
\/p 2 + q 2 + tf
To find the disturbance at Z we have
a + k + h
Ci Cy A . 27r/\t p + u , q + v _ , \
a — k —h
a+k +h
, ( \i Cj A . 27r / Xt p + u q + v rt ~ ,\
■ak
450 Mr. L. N. G. Filon on certain Diffraction Fringes
which, on being integrated, gives
2kbX . 2rrq + Vj . 2rrp + u, . 2ir/Xt \ 2iry
"27 — : \ f , — >sm r  T ™ A: sm — ^— h sin — ( e Icos ,
7r 2 (pju)(q+v) X b \ b \\t J X
where — €y=— V Jl a 2ftcosf, _ 6 + 7= ?; ±£ a _2«cos<£,
whence e = a cos <£ f ft cos >/r,
7= — t— a + p cos >/r — a cos </>.
Hence the intensity I of light at (p, q) is
ir (p + u)*(q + vr X b X b X
, v + 2 Scos 6' — a cos z (ft sin 0' \ a sin 0)
where 7= ^^a + ~ / o b— x a
b x /f + q 2 + b 2 </ p 2 + q * + b * 9
In the last term we may put — y =^ = , for if we went
Vi? 2 + g 2 + b 2 o
to a higher approximation, we should introduce cubes of p/b,
qlb which we have hitherto neglected.
If, further, we make ft cos 6' = a cos 0, which can always he
managed without difficulty, the second term, which would
contain squares on expansion, disappears and we have
(v+ q)a — ( ft sin 0' + asin 6)q
7 _
= \va + q(a(ft sin 0' + a sin 6))\/b.
This gives fringes of breadth bX/2(a— (ft sin & + a sin 0)).
These may be reckoned from the bright fringe 7 = ; i. e.
— va
qQ = a{ftsm6' + *sm0)'
The visibility of the fringes for a single source will, as before,
not be affected by changing a : for a second source the origin
of the fringes is given by
q '=—v'a/(a—^ft sin 0' + asin 0^).
and if the visible rectangles overlap, there will be a sensible
diminution of the fringe appearance whenever
q  q Q ' = (n + i)bX/2\a (ft sin 0' + a sin 0) }
where n is an integer;
i.e. v' — v = an odd multiple of bX/4a,
the condition previously found.
as applied to Micrometric Observations.
451
One further point should be noticed : if a be very nearly
equal to (3 sin & + a sin 6 the fringes become too broad to be
observed, whatever the source may be.
Fig. 5.
zy
T) '
/e\
r
/>/N
sj&'
^K
n
n'
?'
To see the physical meaning of this condition, and also of
the condition ft cos 0' = a cos 0, we notice that a point source
of light P at the centre of one of the slits appears after re
flexion at the two mirrors a, b, to be at/?, where Pp is equal to
twice the distance between the mirrors and is perpendicular
to their plane (fig. 5). Hence the double reflexion removes the
image of the slit a distance 2a cos 6 behind the diaphragm and
2a sin 6 closer to the centre. In the same way the image of
the other slit is brought 2/3 cos 6' behind the diaphragm and
2/3 sin & nearer the centre.
Our condition {3 cos 6' = a cos 6 therefore means that the
images of the two slits must be in the same plane parallel to
the plane of the diaphragm itself, and our second condition
shows that they must be some distance apart.
To find the minimum of this distance, remember that the
fringes will be invisible if the distance between successive
maxima exceeds the vertical dimension of the visible rect
angle : in other words, if
b\/2{a (/3 sin 6' + a sin 6)) > b\/k 9
or distance in question < k,
which means that the centre of the image of either slit must
452 Mr. L. N. G. Filon on certain Diffraction Fringes
be outside the other. These two points must be carefully
borne in mind in adjusting the instruments.
When this,, however, is done, we see that Michelson's
assertions are confirmed, and that when we increase the
aperture of the telescope in this way, the results obtained are
of the same character as when the "slits are placed directly in
front of the objectglass.
5. Let us now proceed to consider an extended source,
which we shall suppose for simplicity to be of uniform
intensity.
The intensity at a point (p, q) on the screen will be of the
form
2irk(q + v) . 27rh(p + u) ^ 2
the integral being taken all over the geometrical image of
the extended source.
We have now three cases to consider.
(a) When the angular dimensions of the source are large
compared with \/h.
(5) When the angular dimensions of the source are small
compared with \/h.
(c) When the angular dimensions of the source are
neither large nor small compared with X/h.
Let us begin with case (a). Then, if we consider a point
inside the geometrical image, the two limits for u will be very
large, except where the vertical through the point cuts the
image ; the quantity
2irli(p + u\
n a
2irh{p + u)
being insensible for all points outside a thin strip (shaded in
the figure) having for its central line the line through p } q
perpendicular to the slits.
We may therefore, in integrating with regard to w, replace
the limits by + oo , and then integrate with regard to v along
the chord of the image perpendicular to the slits.
as applied to Micrometric Observations,
Fig. 6.
453
Hence, remembering that
it follows that
32A 2 M 2
I
sin
■dx=7T,
I
2jrk
(q+v)
b\
2irk (
{ cos 2 jr(q + v)dv.
Now if the angular dimensions of the source of light be
large compared with \/k, the limits of integration with regard
to v may be made infinite. In this case the intensity I
s _ 8AW(
b\ J ao
9 9_
dv {sin 2 — (ahk)(q + v) + sm <2 j^(a — k) {q + v)
b\
bX
< ism * — ( q + v) + 2sm* 1 ^(g + v)}^ (—±—)
4A 2 A
7T
{(a + *) + (a*)— 2a + 2*} ~^^J
= SAVik = constant.
This result shows us that if the dimensions of the source
exceed a certain limit, no diffractionfringes exist at all, at least
near the centre of tbe image. Next let the angular. dimensions
Phil. Mag. S. 5. Vol. 47. No. 2$$. May 1899. 2 I
454 Mr. L. N. Gr. Filon on certain Diffraction Fringes
of the source be less than ^y, then throughout the integration
27rk
~rr~ (.9 + v ) i s ^ ess than t/G numerically.
But
77 9 /36
■i J
and differs but little from unity.
"We may therefore in this case write
. 2 2irk
sm *_ (3+t)) ^
*<*+•>)
V 6X
throughout the range of integration.
If now the limits be v 1 and v 2 we have
i_«?^( W .<S E a) 4 ,
4AW
™ \ TT (ll ~ i3) + sm bx sin bx J
' "4 T^ + 2 sin — \i ^ cos yj \ K J \
ira \_ bX hX bX J
Let 2c = length of chord through the point perpendicular to
the direction of the slits, then
2c = v 1 — v 2 ,
and let v = coordinate of the midpoint of this chord. Then
T 8A 2 M 2 Chirac . kirac krcaiq + v ) 1
I= ^r l^r + sm i?r cos — * r
The fringes therefore disappear when
Azirac
Their visibility is
. 4:7rac 1 4*7rac
as applied to Mierometric Observations. 455
and is a maximum when
4:7rac . \nrac
 w =tan sr j
but the most visible fringes correspond to the early maxima.
This form agrees exactly with the formula given by
Mr. Michelson for a uniformly illuminated segment of a
straight line perpendicular to the slits. We see, however,
that, provided the conditions stated be fulfilled, it is applicable
to a source of any shape.
The most general form of the fringes is given by
9 + 2 ( v i + v 2) — const.,
and therefore consists of lines parallel to the locus of middle
points of chords at right angles to the slits. These will be
straight lines in the case of a rectangular, circular, or elliptic
source. Here, however, a new difficulty presents itself. For
the rectangular source i^ — r 2 will be constant, whatever chord
perpendicular to the slits we may select. Fringes will there
fore appear and disappear as a whole.
But for a circular or elliptic source, v x — v 2 varies as we
pass from chord to chord. Thus the maxima will be invisible
for some chords when they are most visible for others and
conversely. Hence, whatever be the distance between the
slits, it appears at first as if we might always expect a mottled
appearance.
But in the case of a circular or elliptic source the length of
the chord varies extremely slowly near the centre and there
fringes will be visible, the length of the chord being practic
ally constant. The mottled appearance, on the other hand,
will predominate as we approach the sides.
6. Consider now case {b) and let the dimensions of the
source be so small that, for any point sufficiently close to the
centre of the image ^~ is a small angle throughout
the range of integration.
[For points not near the centre of the image the illumina
tion will be very small and the appearances are comparatively
unimportant.]
For a point distant < ^77 from the centre of the image, we
may put, as in previous reasoning,
sin ~&3T^ + t7 ) Sln Zx"( P+M )
" =1
1
2irk , . '2tt1i , N
21 i
456 Mr. L. N. Gr. Filon on certain Diffraction Fringes
all over the range of integration, whence
29A2P7.2
b 2 X
32A 2 M
b 2 \ 2
/~ X.iraq f ±nav 7 . krraq ( . 47r<2w . \
where fl = total area of the image,
R cos c£ = 1 u cos — yr  rfw, Rsin ^> = tw sin . dv,
the integrals being taken all over the image. The visibility
= R/fl and therefore vanishes when R vanishes.
In the case of a circular source we find
<£ = 0, R = (some nonvanishing factor), J } I— — V
where Ji is the Bessel's function of order unity and r is the
radius of the image, so that rjb is the angular radius of the
source. The dark fringes are given by
q being measured from the centre of the source. The fringes
are parallel to the slits and disappear whenever
This result agrees with the one given by Michelson for
any circular source. We see that it only holds provided
the dimensions of the source do not exceed a certain
limit.
In the case of an elliptic source </> = also, and R is not
altered by any sliding of the image parallel to the direction
of the slits. Hence we may replace the oblique ellipse by
one having its principal axis parallel and perpendicular to the
slits, the values of the semiaxes being d and txr, where d —
length of semidiameter of original image parallel to the slits,
tff = length of perpendicular from the centre upon the parallel
tangent. We find without difficulty :
as applied to Micrometric Observations. 457
r, ( 4:7rav
±1 =  cos iidv over the image
47rav <#sr 6\ T (\iratz\
w
The visibility is therefore
7 /47rttOT\ / /A7ravr\
2,h {i^)/\~bir)>
and vanishes whenever
J.^O.
Hence we see that, for an elliptic source, if p = length of semi
diameter perpendicular to the slits, 57 = length of perpendicular
on the tangent parallel to the slits, then the fringes disappear
when sin =0, if the angular dimensions are of order
ja r as indicated above, and when J : I )~0, when the
angular dimensions are less than —  T .
In the first case p is the quantity which determines the
disappearance of the fringes, in the second case w: and
further, we see that the validity of the formulae is entirely
dependent on the length and breadth of the slit, neither of
which is considered by Mr. Michelson.
We may notice that the best results are obtained, in the
first case when h is large, in the second case when h is small.
7. It remains to consider the intermediate case (c). This
does not perhaps present so much interest as the other two ;
the first will generally correspond to the case of a planet, the
second to that of a star, in astronomical observations.
In dealing with case (c) we shall suppose the anoular
dimensions to be small, with regard to \/k 9 but not with
regard to \/h. We may then write
\ «L_ 1 **l_1 cos * **«(*+*) , ,
1 *2irk{q + v) JTrh(p + u) t cos fa <V»
I b\ bX )
. Jlirhu
32A«A 8 ff
IW
I l b\ / ±ira(q + v)\ 1 _
458 Mr. L. K. Gr. Filon on certain Diffraction Fi
if we only consider the appearances along the line /> = 0, taken
to pass through the centre of the image, which is assumed
circular or elliptic.
Denoting yr— by jjl and expanding, we get
sm *^  r1 2(rl)
frX x 2W , , (1) (2H
/2ttA^\ 2 ~ 2 ' 3! + ' * * + *.(2rl)I " t "* ,,?
V bX
whence
32Awf (7 1 (2,0V, flHy 1 *" , )
4™(£+t0 rf 1
X cos
OX
where
taken all over the source, and is essentially positive and in
dependent of a and q.
Now
 w sin — — — tf# =
for a circle or ellipse.
To find the cosine integral, C, we have, d and ot having
the same meaning as before,
C = 4 — 2r i v& 1 — o 2 ) cos —r—dv.
Jo w 6\
Put v = «r cos 0.
C = 4^^J sin* d cos (^cos tf)<tf
T (±ira\
/47T05Y ^ ^ '
T /47TC/'5r\
_ 2fl? a  1 i8nr(2rl) ! A ^ j
2 r ~ 1 (r — 1)! /47raOT\ r
as applied to Micro metric Observations. 459
Hence:
1 =
bW
j 2r l rl /4w7*\ T /47TO^\
bX l^ i / 47raOT / r(2rl) 2*
i (*— 1)1
32A 2 OT. 2
6 2 X 2
TT J_ 9 ^^ ™d ;■ I V 6\ / J\JW
iV + Zcos __ ^ x ti } 1"^^" frl(2f— 1)
Denote by 12 the total area of the image and by e the ratio
/ 47r/^\ 2 //87T«'5T\
Then the visibility
2H 1
T /47ra'sr\ e T /47ra«r\
K 47m*7 i °\Jjr) 6 J VfaT7 +
6\
i
+<«>" ^£r><F) + 
The series for the visibility is absolutely convergent,
because J n +i(®)/Jn{®) decreases numerically without limit
as n increases without limit.
The roots of the equation
/&7ra'&\ e T / liravrX / .. e r ~ l T /Aira'ST\
(Mr)6 J <«r) +  + < 1 )'" l rr(2^r) J {^r) + =
give the values of a for which the visibility vanishes.
Notice, however, that e contains d and the length of the
slit, so that the values obtained for a will be functions of
the horizontal diameter and of the length of the slit.
8. Besides enabling us to determine the angular distance
of two pointsources and the radius of an extended source,
Mr. Michelson's method allows us to detect and measure the
ellipticity of a luminous disk.
Keferringto the formulae for cases (a) and (b), the visibility
vanishes when
sin , " = in case (a),
OA>
and when
ji ' \ = in case (b).
460 On certain Diffraction Fringes.
In either of these cases, if we rotate the slits about the axis
of the telescope, without altering a, then if the source is
elliptic, p and nr will vary, and the visibility of the fringes
will vary.
Now suppose for a given position of the slits we vary a
until the visibility = for that position, and then rotate the
slits and note the different inclinations for which it vanishes.
It will certainty vanish once again before a complete half
turn has been made, namely, when the slits make an angle
with the direction of either axis of the ellipse equal to that
which they made at first, but on the other side of the axis.
It may vanish more than once, but since the inclinations
for which it vanishes are symmetrical with regard to the
axes of the ellipse, there will usually be no difficulty in deter
mining the directions of the axes.
Their lengths can then be determined by two observations
of the disappearance of the fringes, one for each of the two
positions of the slits which are perpendicular to an axis.
It must, however, be noticed that the accuracy of this
method for measuring ellipticity decreases with the size of
the source, inasmuch as the quantity which causes the altera
tion in the fringes is the difference, not the ratio of the semi
axes.
To get some idea of the sensitiveness of the method, let us
estimate roughly the amount of ellipticity which could be
detected in a disk of angular semidiameter 10 7 ', taking the
mean wavelength of light "5 X 10 4 cms.
The visibility vanishes when sin = 0, and will be
quite sensible when sin , " = ^, say. Hence in order that
we may be able to note a sensible difference of visibility in
the fringes, we must have
£(*?); le^;
or
P1—P2 _ 1_ JL
b "2 10 6
if a be a little above 4 cms.
.*. difference of angular semiaxes ='01 (semidiameter)
g.p.j or the amount of ellipticity which can be detected =*01.
I have taken sin—  —  = as giving zero visibility, because
this example will clearly fall under case (a).
On the Absorption of Water in Hot Glass. 461
9. Summing up the results obtained we see that :—
(1) It is possible by the observation of Michelson's interfer
encefringes to separate a double pointsource, or detect
breadth and ellipticity in a slightly extended source.
(2) But the distance between the two points, or the dimen
sions of the extended source, must lie within certain limits
depending on the length and breadth of the slits *.
(3) The dimensions of the slits also considerably affect the
general theory, the formulae obtained not being identical with
Michelson's. The law of appearance and disappearance of
the fringes depends very largely on the distance between
the points or the dimensions of the extended source.
XLII. The Absorption of Water in Hot Glass. Second Paper.
By Carl Barus f.
1. A FTER finishing my account X of the action of hot
a\. water on glass, observed in fine  bore capillary
tubes, it seemed to me that the experiments made several
years ago (1891) left questions of considerable interest out
standing. I refer in particular to a further examination of
the contents of the capillary tubes. Certain evidence was to
be obtained as to the occurrence of syrupy glass at 185°,
solidifying to a firm glass when cold, the composition
remaining unchanged except as to the water absorbed.
Again, as it was improbable that volumecontraction would
continue at the same rate indefinitely, the conditions of sub
sidence were to be determined, together with the effect of the
elastic and viscous constants of the tube itself on the apparent
volume contractions and compressibilities observed. Ques
tions relative to the acceleration of the reaction at higher
temperatures were to be held in view.
Finally, if the inferences drawn from data for capillary
tubes are correct, it should be possible to obtain the fusible
glass on a larger scale. Experiments in progress in this
direction have proved quite successful, and will be described
in a later paper. The present remarks are restricted to the
experiences with capillary tubes.
2. The apparatus used was the same as that heretofore
* Since the above was read, a paper has appeared in the Comptes
Rendus de VAcademie des Sciences for Nov. 28, 1898, dealing with the
modifications in Michelson's formulae when we take into account the
breadth of the slits. The author, M. Hatny, follows Michelson in not
considering variations of intensity parallel to the slits. This, I think,
accounts for his results not quite agreeing with mine.
+ Communicated by the Author.
X Phil. Mag. (5) xlvii. p. 104, Jan. 1899.
462 Dr. Carl Barus on the Absorption of
described (I. c), a clear thread of water being enclosed in a
stout capillary tube between terminal threads of mercury, the
upper of which was sealed in place, while the lower trans
mitted the pressure of a forcepump. The motion of each
meniscus was observed in the lapse of time through a clear
boiling tube (vapourbath, 185° to 210°) with the catheto
meter.
The progress of the experiments may be described, as a
whole, as follows : — During the first stages of heating the
clear thread of water expands, inasmuch as the constant
temperature in question is being approached. After this an
initially rapid contraction of the thread is manifest, which
must have begun much before the period of constant tem
perature w T as reached, so that the full threadlength for 210°
is never quite attained. During the early and most marked
period* of contraction (and some time before) the tube
appears white and opaque, and the observer can only with
difficulty follow the rise of the lower mercury meniscus. The
top meniscus remains in place. Compressibility is a rapidly
increasing quantity. During the later stages f of heating
the tube becomes transparent again, the mercurythreads
stand out brilliantly, and the whitish opaque matter gradually
vanishes in the axis of the tube. Contraction becomes less
marked and finally ceases ; and with it the accentuated com
pressibility of the aqueous silicate, now so thickly viscid as to
retain cavities, also disappears. During this second stage,
threads of mercury invariably break off if there is change of
pressure. Nevertheless, measurement by means of these
indices is not impossible, and in the telescope the observer
notices a slow advance of the viscous mass, moving as a whole
continually towards the upper end of the tube and carrying
the little mercury globules along like debris in a common
current.
To measure compressibility at this stage is to face a
dilemma : on increasing pressure from below, there is marked
increase in the upward motion of the viscous current. It is
difficult to state when this accentuated motion ceases. On
removing pressure the mercury does not retreat proportion
ately, if at all. However, when pressure is reduced too far,
the mercury may retreat several centimetres, quite out of the
field of view, as a whole, leaving well rounded or ovoid
cavities behind. Thus it is impossible to make measurements
* Undissolved glass coagolum.
t Dissolving coaffulum
Water in Hot Glass. 463
for compressibility in triplets, and the data are given below
with, this reservation. These data accentuate the absence of
effective volume elasticity under the conditions stated.
When the viscous thread has appreciably ceased to contract
(12 hours), and the tube is allowed to cool very carefully,
bubbles make their appearance very much resembling those
in a Prince Rupert drop, and probably due to a similar
cause *. They begin to form in the axis, and are usually
connected by a fine channel. They may grow to a diameter
of over  the width of the bore. The formation of these
bubbles on cooling is proof that the aqueous silicate is still
liquid at the temperature of the vapourbath (185°210°),
however viscous and incompressible it may have become.
The solidifying core of waterglass contracts from the centre
outward, and must contract more rapidly than the igneous
glass envelope. It is this last stage (contents cleared again
to a pellucid jelly) which I failed to fully observe in my
earlier experiments, believing that solidification had set in
when the mercurythreads broke off.
After the tube has passed the final stage with subsidence of
contraction, it invariably breaks throughout its length when
cold, in such a way as at first sight to suggest expansion on
solidification of the aqueous silicate within. It makes no
difference how carefully the cooling is performed. If a thread
of fusible metal is allowed to solidify in a capillary tube, the
latter breaks sooner or later in the same way. 1 do not by
any means imply that the aqueous silicate does actually
expand on solidifying, for there are other and better ways in
which the breakage can be explained. The appearance of
bubbles, for instance, is evidence of contraction, and the
breakage is rather due to an excessively shrinking core.
When the cold tube is cut across, the core of waterglass
practically fills the tube, and to all appearances is as hard,
clear, and firm as the igneous glass surrounding it. There is
a difference of refraction between the two glasses sufficient to
make the aqueous core apparent under favourable illumination;
but for this and the bubbles, the tube would be undistin
guishable from a glass rod.
If, however, the end of the glass tube is slightly heated
above a candleflame or a small bunsenfiame, the core soon
begins to melt, to swell up enormously in bulk as the result
of the frothing which accompanies the escape of steam. The
* It is to be remembered, however, that whereas the Rupert drop is
cooled down from above 1000° C, the present high temperature is only
about 200°.
46*4 Dr. Carl Barus on the, Absorption of
result is a light, very white pumice, larging exuding from
the capillary canal (as shown in fig. 1), the part remaining
within resembling pith. In this way the enlarged bore of
the tube may be clearly compared with the parts left free
'88 mm.
Omm.
Fig. 1. — Capillar/ tube, with waterglass core heated above.
Figs. 213. — Diagrams. The crosshatched parts denote the core of
waterglass (much enlarged), the envelope of igneous glass being
ignored. Mercury is shown in black ; cavities in white ; residual
water (?) is differently crosshatched. The actual contours were
usually more ovoid than these figures.
from water and in contact with mercury only. On further
fusion the pumice melts igneously, and, if the glass contains
lead, it turns black in the usual way on reduction.
Water in Hot Glass. 465
There is good reason to suppose that the aqueous silicate
remains homogeneous from the time of incipient solution to
the eventual occurrence of a viscous glassy coagulate, liquid
enough at 200° to admit of the formation of internal bubbles
on cooling and contraction, but at the same time viscous
enough to keep similar bubbles in shape and position without
cooling. A thin thread of water (01 to *01 centim. in diam.)
is undergoing lateral diffusion into the glass, and concentra
tion difference is virtually confined to the cylindrical surface
of contact between igneous glass and waterglass, widening
as the action goes on so slowly, that the much more liquid
waterglass is free to remain homogeneous. The latter
should therefore be identical in composition with the original
glass, but for the incorporated water.
In a final experiment (tube No. 7) 1 put a solution of
cobalt nitrate into the tube, rather with the expectation of
finding blue waterglass as the result. No such action
occurred ; instead of it, the water diffused into the glass as
usual and the cobaltic nitrate was left as a granular scum in
the axis. Chemical decomposition and incorporation of the
cobalt did not therefore occur at 200°.
3. Instead of tabulating the large number of observations
made (in all seven tubes were employed), it will conduce to
clearness to present the work graphically. In so doing, the
data for the former tubes may be included, for reference.
Time is laid off in intervals of 20 minutes (between vertical
lines) along the abscissas, while the ordinates indicate the
changes of length of the column of water in intervals of
1 centim. It is not convenient to specify the full length of
the thread in the chart ; but a datum for the length of the
cold thread at about 20 c Cwill be given both in the latter and
in the tables containing original and final bore, and similar
specifications for each capillary tube. These diameters were
measured with the cathetometer, the tubes being cut across
and looked at endwise. Slight heating increased the sharp
ness of definition between the original glass and the solid
core of waterglass within.
The chart also contains the mean compressibilities j3, for
pressure intervals of about 100 atm., no attempt to obtain ft
as a function of pressure being made for the reasons stated in
§ 2. Since /3=(v/Y)/p, or the decrement of volume per
unit of volume per atmosphere, or practically decrement of
length per centim. of column per atmosphere, the abscissas as
above are successive timeintervals of 10 minutes each, while
the scale of ordinates is a change of /3 of 000100, Several
466
Dr. Carl Bams on the Absorption of
/3values are usually attached to the compressibilitycurves for
orientation.
Calling the two capillary tubes formerly used (preceding
paper, 1. c.) Nos. 1 and 2, their dimensional constants before
and after corrosion were as follows : —
No. 1
No. 2
Internal diameter
(centim.).
Section.
Cold thread
length.
184 em. at 28°
140 cm. at 24°
Original.
Corroded.
Original. 1 Corroded.
•043
■<m
•052
•034
•0015 0021
•0014 0023

In these cases the temperature of the vapourbath was but
1 85°, and the corrosion did not outrun the opaque stage. Seen
under the microscope, the solid waterglass nearly filled the
bore, being an opalescent warty accretion. The chart (p. 470)
contains the results for decrement of length of thread and of
compressibility so far as observed, after constant temperature
had set in. In all these experiments /3 is thoroughly deter
mined * from at least four measurements between 20 and
400 atm., the initial lengths of thread returning on removal
of pressure.
In case of No. 1, time was not observed until after nearly
an hour's boiling. The short curves thus refer to the end of
the experiments. The general conclusions are stated in the
preceding paper.
4. The first of the new tubes to be heated was No. 3, with
the following constants : —
Temperature of Vapourhath, 185°.
Internal diameter (cm.).
Original 0295
Corroded
•041;
Section.
•00068
•00135
Cold threadlength
174 cm. at 23°.
External diameter
about "3 cm.
The observations with this tube were not satisfactory ;
possibly the temperature of the vapourbath was insufficiently
constant ; possibly the clear stage of reaction is not reached
at 185° (cf. §12). The tube turned opaque and so remained.
The fouled meniscus was frequently lost, or could not be
* Cf. Am. Journ. Sci. xli. p. 110 (1891).
Water in Hot Glass. 467
recognized with certainty. No observations were possible
until after 30 minutes' boiling.
In the absence of systematic data, however, certain inci
dental results of interest were obtained with this tube, by
observing at high and low temperatures alternately. Thus,
after about an hour's boiling, /3 fell off from 120/10 6 at 185°
to about 40/10 6 at ordinary temperatures. After about
2i hours' boiling, £ fell off from about 230/10 6 at 185° to
90/10 6 at ordinary temperatures. After about four hours'
boiling the hot compressibilities reached 400/10 6 ; but the
tube broke before the corresponding cold compressibilities
were measured.
Opportunity was also afforded for roughly measuring the
coefficients of expansion of waterglass. After two and a
half hours' boiling the mean coefficient per degree was
about '002, between 25° and 185° 0. The corresponding
mean coefficient for pure water, according to Mendeleef *, is
•0008 to 0009.
Thus it appears that even the relative contraction of water
glass is over twice as large as the normal value for water
under like conditions. After four hours' boiling another
measurement of the coefficient of expansion of waterglass
gave nearly the same result as before. This coefficient did
not, therefore, keep pace with the increase of j3 ; but, as only
a small part of the thermal contraction on cooling lies in the
region of high temperature to which /3 refers, the two
coefficients should not keep pace.
Finally, a comparison between /3 and volumecontraction
in the lapse of time showed a change of /9 of about 18/10 G
for each per cent, of volumecontraction relative to the cold
volume at 20° C. This mean rate is more rapid than the
rate observed for tubes 1 and 2, or the following tubes.
5. The next tube examined was No. 4, with the following
constants : —
Temperature of Vapourbath, 210°.
Internal diameter (cm.)
Section.
Original 0240
•00045
Cold threadlength
1543 cm. at 20°.
Corroded .... 0710
•00396
External diameter,
•55 cm.
A higher temperature of exposure (210°) was here chosen
for comparison, in order t